Episode

My guest today is Luigi Acerbi, Associate Professor of Artificial Human Intelligence at the University of Helsinki, where he leads the Machine and Human Intelligence Lab.

I've been following Luigi's work for years, and it sits right next to a bunch of episodes we've done recently.

Luigi is, above all, a tool builder.

He came to Bayesian inference from theoretical physics and computational neuroscience.

Chasing one idea that inference and prediction might be the thing that brains, science, and machines all have in common.

Out of that came methods like VBMC and BADS and the Pi VBMC and PyBADS packages that a lot of scientists actually use.

More recently, Luigi has fallen in love with neural processes, and we get into why transformers turn out to be such a natural fit.

them.

We talk about his amortized conditioning engine, active data acquisition, changing priors at runtime, and where all of this is heading.

Foundation models for inference.

We even get into small local LLMs and Bayesian decision theory for agents.

This is Learning Bayesian Statistics, episode 161, recorded May 27, 2026.

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Luigi Acerbi, benvenuto, I learning Basion Statistics.

Yeah, I I would love to do the the episode in in Italian.

Um my Italian is very, very rusty.

I I used to speak pretty pretty good Italian, but then I went to Argentina and had to learn Spanish and uh the two languages are so close that um like it just like my brain

just uh confuses it.

So I need I guess I need to go live in Italy again.

Sure.

Yes.

I mean actually I noticed that my my wife is Spanish, so let's say so we ah we kinda we we speak English.

The official language is English in the house, but we we kinda you know, so we pick up the each other's language, so let's say so I speak a bit of Spanish and speaks uh actually she

speaks way better Italian than than well than me than my Spanish.

Anyways.

Yeah, okay.

yeah, I mean that's now I'm

The weird thing is that so now when I come to Italy, my Italian comes back after a week, more or less, but the first week I'm like the I'm the classic proverbial French that

people hate uh who thinks you know they who think they speak Italian but they actually speak Spanish.

And and I know Italians hate that and I don't do it on purpose, so okay.

I'm I'm sorry, I don't mind.

I don't mind.

It it's close enough.

Awesome.

Well really, really happy to to have you on the show because I've been following your work for a few years now and you do super interesting stuff related to to a few episodes we've

done now on the show, especially with um with Marvin Schmitt and uh even more recently with uh Stefan Kalef.

Episodes one hundred seven, one hundred fifty seven, one fifty eight.

These are gonna be in the related uh episodes for this one.

But so for today, uh let's start as usual with your origin story, Luigi.

What uh what are you doing nowadays and how do you end up there?

All right, yeah.

Well thanks thanks for having me.

So where am I?

So well we're now physically I'm in Helsinki and um so my professor, associate professor of artificial human intelligence.

Uh

The work we've been doing uh recently in the past few years focuses on using modern machine learning tools to do Bayesian inference.

That's that's kind of the broader area.

And uh we apply these tools sort of to various uh domains.

One goal is to do just predictions and inference.

but let's say one application is obviously uh model fitting in the sense that you have a model you want to

you know, infer the parameters of that model.

Uh, you want to make predictions with that model.

And nowadays, let's say there are powerful you know neural networks that allow you to kind of do that uh fast and anyways that that's the direction that we are uh following

recently.

But how I got here is a much longer path.

And so I'll try to compress it, like literally how you know modern agents when you do compaction, so I'll try to compact

a very long string of tokens into a few sentences.

Originally say as we were discussing a while like a little ago, uh yeah, so I mean I grew up and studied in Italy, uh but physics, that that's kind of where I come from.

theoretical physics.

But then uh for my PhD I moved to the UK, Edinburgh, and where I studied uh was researching computational neuroscience.

Uh at the time it was called computational neuroscience and neuroinformatics.

And

The brain is very confusing, uh, especially if you don't know much biology, and I didn't definitely know enough biology at the time.

No now.

And um neuroscience is even more confusing because it's a also I mean it's very interesting, but it's also very, very confusing.

And as a physicist, always, you know, one of the things that teach you is that you know you want to simplify things as much as possible.

So let's say and trying to simplify the brain is hard.

Uh possibly let's say you know, there are o also let's say

disciplines or currents that say that you shouldn't try to simplify too much.

But, anyways, but you know, that's kind of the temptation.

You want to kind of come to the simplest possible description in a way.

And uh I was kind of failing to try and find to make trying to make sense of it.

And then I read a few influential things and I studied a few influential things.

uh One first I think I say one one one book that kind of influenced me at the time and actually convinced me to go to computational neuroscience was uh On Intelligence.

And and the idea that

uh what the brain is doing is prediction and that seemed like a beautiful so everything that the brain is doing everything like in the cortical columns in the brain are doing is

just you know kind of predicting uh predicting inputs and I found that you know a very powerful idea at the time uh and then you I studied well studied slash started researching

in computational neuroscience and then I was exposed exposed at the time to Bayes and uh

It that was in a curse.

Uh as a physicist, I had absolutely no idea of what Bayes was, because it's a mm that's not taught.

It wasn't taught at the time.

And so let's say I just had, you know, some kind of sprinkling of you know classical statistics.

And then you know, when I was kind of exposed to Bayes theorem, it's like, okay, okay, this explains everything.

And um and then essentially not just Bayes, that that just was based on in a machine learning course, uh probably still machine learning.

But the and then let's say the idea that the the brain

does based on inference and what the brain, you know, essentially what it does in a nutshell, everything that the brain does, I mean not everything, but say a a big chunk of

what the brain does is being a sort of statistical inference engine.

It's kind of you know trying to make sense of the external world by building model, you know, assigning probabilities, you know, computing posteriors.

And I find that idea uh very alluring.

And then essentially then we that essentially that's kind of everything kind of came together.

So it's just starting to study this version of

kind new science con live science.

Then of course then I became more practical with the tools of Bayesian inference.

But then again, I was kind of studying Bayesian inference because the system I was studying was using Bayesian inference.

So I was kind of trying to start a system, you know.

But then you know you can use the tools of Bayesian inference to study the system which uses Bayesian inference.

So I was doing kind of sort of a Bayesian inception in which you know I was using Bayesian tools or Bayesian models to understand a Bayesian system.

And

People have called this a doubly Bayesian.

I argue that what I was doing then became tri triply Bayesian because it's say then I kind of I started using tools which themselves used Bayesian inference to perform their task,

like and I will get to to those later.

So anyway, so I had base everywhere in my life.

And uh and then essentially so I started building tools to do model fitting.

And then I kind of like that part.

Uh these models are complicated.

Uh requires you know complicated tools to fit the parameters.

And and then essentially I started

developing those tools as part of my job.

Because anyways, Jammer is the same.

You have base and you know whether, you know, how you apply it is up to you.

So everything became essentially yes, you know, building tools.

And then eventually I kind of moved even more into machine learning because again, as as I tend to say, I'm interested effectively, you know, again, in this um this idea that

inference is kind of kind of some sort of a foundational, fundamental uh concept

in the universe or uh you know that's uh that's useful you know say in in in biology is useful in in in science and so so I'm studying it you know in all these different areas.

And then so essentially so it was kind of natural to move to machine learning to do you know the the the things we are doing now.

Uh because again uh studying you know say the the the end system in a way is relatively relevant.

What what kind of the

the core element that stays the same is exactly this this idea of uh inference and prediction.

Yeah.

Yeah.

Um makes sense.

And it's really interesting to have the to hear you about the the the you know the the whole arc of your your career.

Um so you've touched on it a bit, but maybe um even more specifically, what if is there anything in particular or someone in particular that drew you to patient statistics

specifically?

And and then to augmentise the inference and and neural processes?

So the path so yes, so let's say so I had a few uh professors at uh when I was still let's say a master student and a PhD student in Edinburgh.

Um at the time there was I mean let's say still there is uh a very, very vivid and interactive

machine learning and computer science and also at the time say a neuroscience, computational neuroscience community.

And uh so there was a lot of interconnection between uh our cohort of computational neuroscientists and the machine learning cohort.

In fact let's say people were kind of swapping all the time.

The I think I said there was one course which I took at the time, which was uh something called like

probably something like probabilistic quality modeling, which is similar to the course I'm currently giving.

And and so yeah that introduced essentially among other things introduced um the idea of the Bayesian brain or the idea that you know the brain does probabilistic inference.

At the time it was taught by uh Peggy Series.

she was uh lecturer at the University of anywhere at the time.

And so yeah so that was quite inspiring and then kinda I I took uh that route and then I I kinda you know never really

left it.

And uh coming to uh today, you were asking about well, you know, what I'm doing today, you're asking about the neural processes.

So actually this is quite kind of funny because neural processes are essentially neural networks that uh learn to do inferential and prediction tasks.

And I stayed away on purpose from a neural network for most of my life.

That was a conscious choice, possibly wrong, actually surely wrong.

And

And the reason is that let's say when I when I was a PhD student at the time, let's say neural network deep deep nest, well actually at the beginning they didn't really exist.

Deep next exploded, let's say, in 2000, you know, famously people mentioned, you know, AlexNet at 2012.

Um and so as the year in which you know deep learning started working.

And I started my PhD in 2010.

So let's say so when I started it was it wasn't quite working yet.

And uh but still let's say at the time for a number of reasons, let's say I wasn't interested in neural networks.

And that was a mistake, obviously, uh in hindsight.

but so I say so so I kind of over took me a while to kind of get back to neural network.

Let's say I did it very late in my career, like I did it like probably a few years ago.

And what exactly what kind of convinced me to come back were neural processes.

so neural processes, so there were these beautiful papers in twenty eighteen.

I kind of discovered them maybe a couple of years later, one two years later.

And these so the the first author Marta Garnelo, I think.

I think she's at the time she was a deep mind, uh, I think.

Uh now I'm not sure.

But let's say, and you know, this was yeah, so anyway, so uh group at uh deep mind, if I'm not mistaken.

And uh the idea of the original neural processes paper, there are kind of two papers.

One is called neural processes, the other conditional neural processes, but yeah, they're they're similar in in scope.

The idea is that you can train a neural network.

At the time it was kind of like a standard uh multilayer perception, so like you know, like relatively simple uh

network.

So normally how we train a network is that uh you have a data set and you know you train on a single data set.

I mean that's people how people you used to do it.

So you would train on a single data set.

Maybe you have a held out data set that you use for you know validation.

But you you you train on a data set and then you have to perform a task and you have a single data set that you train on.

The idea that they had at the time is is that instead of training on a single data set, you train on a data set of data sets.

So say you have a lot of you know kind of small data sets that you're training on.

And for each one of them, you're training your network given the small data set uh to do a task like you know predicting uh, for example, the val you know giving us a small uh data

set of points, like you know, an aggression problem, I give you x and y value for each.

So that's my data set.

Then you know I want to predict you know the value of the function at other points, other values of x.

So what you do, you take your data set, you split it in a a training set and a test set, and in the in the

neural process literature you will call them context and targets.

And by the way, context will come back.

And um and you know, and the idea is that okay, so now I'm training my neural network to given the context to predict the targets.

You do it once, twice, one thousand times, because you know you have this a lot of small data set that you can chop, you know, uh you chop in context and target and you repeat

this procedure a lot.

So now what you're doing is you're you're training your network that, you know,

To do this task, what you do this prediction predictive task, and then you know, at runtime, now you give it another data set, which you call context, and you give it a bunch

of targets to predict, and your network has learned to solve this prediction task.

Okay.

So but the point is, of course, it's done it by it's done it by learning, by seeing a lot of different data sets.

And of course, I say the the idea is that you would use um, for example, some generative process.

or a data set of data sets to train your network in such a way that it kind of learned the statistics of the task such that when you present a new instance, it can just resolve it

immediately.

And nowadays, or already at the time, but you nowadays, you know, we would call this a form of amortized inference in the sense that you can do this training on many, many, many

data data sets, and then you know you just deploy it and then at runtime you can do your predictions on new data points, you know, kind of instantly.

And I found it it was beautiful because it's a, for example, when you condition on a bunch of points, that's a set, typically.

So it's I mean, unless it's time series, but it's a you know typically it's a set.

that's you know also a property.

So kind of you want to satisfy some prob some properties of probability of stochastic processes, like for example, you know, if you if you change the order of the points, you

know, your prediction shouldn't change because it's a set, the data points are exchangeable.

Uh if you say that's the term the term.

And they they use a structure which is called a deep set to represent this set with a neural network.

And so essentially so it's a mix of you know Bayesian statistics, you know, stochastic processes like Gaussian processes, for example.

Um, but let's say anyone uses deep networks, but let's say it's kind of also mathematically beautiful because let's say you have you know you have these properties

that gets you know uh so which are preserved.

These are these the can the fact that you can move you know permute your

your nice small set of data.

So anyways, and then you so you have something that learns a task.

And I found that actually really, really interesting and promising and kind of, you know, very elegant in a way, as as opposed to maybe, you know, kind of just, okay, I I train a

network to do a bunch of stuff.

Like, you know, this is a there is some elegance to the to the concept of of a new process.

And I fell in love with that.

And then you know I started digging and you know and then we get to today.

Super cool.

Yeah, yeah.

Damn.

Thanks for uh thanks for all these details and

And actually so one of the things you you've been working a lot is um is a concept we haven't touched down yet really on the show, which is called VBMC for variational patient

Monte Carlo.

And so you describe it as like patient optimization, but for finding the full posterior instead of just the optimum.

Wha what does that mean?

What is the core

frequency in the core method that makes that possible.

Okay.

Cool.

Okay.

So yes.

So to connect a bit to what they just said.

So before I kind of mention in pass in passing Gaussian processes.

So I said so that was kind of my intermediate phase, which is still ongoing, let's say.

So so before I move to kind of I I came back to neural networks.

Actually I mean not I came back.

I I came to neural networks.

So I was kind of still in the camp of kind of traditional Bayesian non-parametrics.

which is users cando, yeah, and and the the workers of that field is the Gaussian processes.

And uh so I mean I may I will assume that you know your audience might have heard of of Bayesian optimization, which is the idea is that you have uh essentially you want to

optimize a function.

Uh this function might be expensive.

so let's say so you know you have a a a limited number of evaluations you can take to find the optimum of this function.

So what you would do say cando traditionally

Is that uh one approach is called is called Bayesian optimization.

And the idea is that you take your your evaluations, you can use some form of uh of regression or surrogate model that you're fitting to these points, and then you use this

surrogate to decide okay where to sample next.

And and the goal of this uh of this procedure is that of course that you want to find points which are promising in the sense that they tell you where the optimum might be.

And they kind of guide your sampling procedure so that you know you can find, you know, a good candidate for for the optimum.

And yeah, the the as I said, the workers for this is a Gaussian process, which is again a Bayesian kind of mathematical object, which has a lot of nice, very, very nice and pretty

properties.

Everybody kind of loves Gaussian processes when they can use them because they're really, really mathematically beautiful.

Uh they have a bunch of kind of downsides, but otherwise I'd say yeah, they're they're they're really practical and

They have again so one of the example some of the properties they have is that they they behave very well with with little data.

So let's say, because essentially they're they're they're a Bayesian kind of construct.

So they come up with a strong prior for what uh a function looks like.

For example, your function depending on some choices you made, but typically you assume the functions are so much smooth.

So let's say so that your function is not going to do some something crazy, but it's kind of you know, it interpolates between points nicely.

And uh so people use it a lot in uh in optimization.

And that's kind of Bayesian optimization is kind of it's a it's a field in itself, it's a big field.

And there is a beautiful book by Roman Garnett recently came out, Bayesian Optimization.

And you know, I I really recommend it's it's a really, really awesome book.

And anyways, so that's optimization.

But I'm Beijing, I don't want to just optimize, I want to find the full posterior.

And so let's say so.

The idea I had uh many years ago uh was well.

Can so because you know when when you're building your your third model using a Gaussian process, you're kind of representing a function.

And of course at the end you only care about the optimum, but say, no, you're kind of representing the shape.

So can we use that?

But you know, where the function is not an arbitrary function, but it's it's effectively a posterior, and more specifically the a log posterior, so the logarithm of the posterior,

and you know, it's it's a function.

So can I use something similar to Bayesian optimization to kind of you know sample

This distribution.

By sample when when I mean when I say sample, I mean evaluate at arbitrary point, not sample like as in MCMC.

So you know, so I'm just you know evaluating this function at a few points.

And then you know I use this very powerful interpolator to kind of build kind of you know the entire shape.

And then instead of optimizing it, what I want to do with this shape is no no, I I just I keep all the shape and that's the posterior.

So I want to learn this shape of the posterior by a few carefully selected points.

The reason to do that.

Of course to say if your function like if if you say if you're working with a model that you can write in in Stan or Py C, probably that's not very expensive, so you can run C C.

But you know, there are lots of models in in main disciplines where you can write um, you know, your your model, your log likelihood, but evaluating it is expensive.

So let's say so you can calculate it, but I say it's expensive.

So I say, so you so

And you and that's essentially the the idea of VBMC that essentially kind of does kind of the same job as basic optimization.

So it lets you get the shape of the function.

But oh instead of you know being expensive and taking, you know, MCMC might take you know tens of thousands, hundreds of thousands, even million evaluation of your function, which

is your function even takes one second per evaluation.

You know, you need to wait one million second to to do a full MCMC.

While VBMC typically, you know, with

a few hundred evaluations, you can, you know, kind of get a a good reconstruction of the shape of the posterior.

It's not perfect, uh you know, we can discuss it.

But let's say so it's again, it's a very good approximation typically in low dimension.

That's that's the cave the essentially the big caveat is that you know this works well in low dimension because you're building this surrogate and as anybody, you know, you work

with GPs, production processes, um yeah, they work well, you know, in low wish dimension.

Let's say, you know, if you b go above, you know, ten, twenty, thirty, let's say it's kind of gets hard

Just you you don't have enough point to kind of interpolate uh the shape of the function.

So it works for models that have up to maybe five, ten, fifteen parameters.

Damn.

Okay.

Yeah.

So that means how would you in which cases would you use that concretely?

I think it's gonna be interesting for for listeners to understand.

and

While preparing for the show, I was actually looking at your PyVBMC package, which looks super cool.

I I encourage people to to go in there and and give it a try.

so actually can you can you tell us when should we reach for for that package?

When is it not appropriate?

I think it will give us an idea of when these kind of of methods are most powerful.

Yeah, yeah, thank that's that's a good point.

So so one thing I want to say that so I always the reason why I moved to to kind of doing these things exactly because I came from the practitioner side.

So I was building models and fitting models and it was slow.

So let's say so I say, okay, I had an actual problem I wanted to solve.

And many of the users of VBMC or of the other our other tools like BADS are are practitioners.

They they work in cognitive science, computational neuroscience, and now you know it's kind of spreading to other fields.

You know, kind of but let's say so.

So yeah, so if you if you look, for example, at the at the website, there is always, you know, I've I've big FAQs that are gonna tell you, you know, how when it works, when it

doesn't, because exactly we kind of want to uh be clear for the user and say and you need to be very honest about what what a method does and and doesn't.

So in the case of PyVBMC, so when should you wish it out?

First of all, let's say so if you're if you can code up your model in in Stan, in in PyMC, um you know, everything is working fine.

you're good.

So you know there is no need for for prime MNC.

so the point is, you know, whet say if you know that you say if you have a problem and you know you have a problem, you know, in the sense that okay, so for example, you know, you

have a model which for a number of reasons, doesn't you know, y sometimes you can't even code it in a in um say in Py C or in uh or in Stan because say there are some aspects

which are non differentiable.

So let's say sorry in trouble because you you say

These methods would need to compute the gradients, I mean, automatically to do MCMC.

So for example, you know, if you're for for a number of reasons, your your function is not an easy function, but say there's some sort of a complex process that you need to solve.

And these emerge in um in the computational sciences.

So for example, you know, if you're building a statistical model from scratch, this is rare because say by definition, you you think in terms of easy, nice distribution that you

can fit.

But let's say uh if you come from the science, it's kind of the opposite in the sense that you know you don't think in terms of statistics or distribution pack, you know, you think

in terms of the process that you're trying to model, and that might not kind of behave like any of our nice distribution that you can get out of a package.

Let's say so strategy.

So I would say most of the usages come from science and engineering, where you're kind of trying to model something which is kind of has its own process that you're trying to

model, and then you know, say then

And the the kind of symptom that you have, you know, it's not that you know the process complex or not, but the symptom that you have in practice is that uh, for example, it

takes time to compute.

So I say so just you know, evaluating one uh point, one param, you the log likelihood or log density at one point, you know, takes you know, the order maybe you know, one second

or something.

One second might not seem a lot, but I say it is a lot, because again, if you need to sample like if you evaluate your function in again, tens of thousands of points, yeah, uh

that's you start feeling it.

So say speed typically is kind of the first symptom that you know people get aware of.

And this might happen, let's say, even if you have a classical kind of easy model that you could fit with um you know, with Stan, et cetera.

But maybe for some reason, let's say either you you have a lot of data or there is something that makes this particularly nasty.

Uh and then let's say, so I can maybe the the time factor.

So I would say the time factor is the kind of the first one that people would would notice.

So if you're in that case.

Then there are a few kind of other check uh check marks that you need to check.

Because for example, since we use this Gaussian process as as a surrogate, and you know, so essentially you're doing interpolation in well, essentially you're doing interpolation

in this in these spaces.

Um so we we kind of know, let's say, from you know, from we using Gaussian processing regression, invasion optimization.

So these things work well.

They up to order of ten parameters or

Which means you know maybe you can push them a bit further, you can maybe get to fifteen, et cetera.

But I say so if you're for example, you if your model has you know fifty parameters, I would not recommend uh Pi BMC in its current form at least.

You know, who knows, maybe you know, we'll we'll we'll push it to high dimensional.

but again, it's in again, it's like other problems in uh in basic optimization.

The so there is a so why, for example, there is that limit?

Because in base optimization in the end, you know, using the surrogate just to kind of to guide you to find a good point.

But then you then you're evaluating your your function.

So optimization is is an easy problem because it's self-verified in the sense that you can evaluate a point, you know, you measure it, and you can just check whether it is a a

better point than the previous one.

Then you know you know whether you know you're kind of you're improving your optimization.

It's kind of self uh you know um verified.

That's not true for Beijon for the Beijing inference, because say you're you have a distribution and it's absolutely non trivial to diagnose

whether you know your distribution is improving.

What does it even mean?

Let's say so you just get a different distribution.

But let's say there is no there is no immediate ground truth that tells you whether something is correct or not.

So anyways, so it's a way harder problem.

So let's say so there's a lot of things that VBMC does, PyVBMC does, um to make sure that inference is going correctly.

So let's say there's a bunch of diagnostics.

but anyways, so let's say so this this is check check the speed of your evaluations.

And check the dimensions.

These are kind of the starting point.

Then you know there are other things that you can start looking.

But I say so.

If you are in this case, then you know, then it might be useful.

Yeah, okay.

Very practical.

Thanks.

And and that's indeed what I really love from your research is that it's not ivory tower research.

It's uh it's really okay, I have this problem, how do I solve that?

Not only for me, but for uh for everybody.

So so I think I think that's what makes what you're doing and and also what the whole group at

Helsinki University and Aalto mm is making is just always very, very practical and helpful.

And so accompanying these papers with uh with open source packages as you do is is always a great way to make sure papers actually go through to practitioners' workflows, which is

in the end what I'm guessing is the most useful.

Um and so you have you have another

package actually that's called PyBADS.

Can you tell us what the difference is?

Like what is BADS in comparison to V B C and and when is that helpful?

You know, which cases uh does it help?

Yeah.

Yeah thanks.

By the way I didn't mention what VBMC stands for.

That's it's it's it's a mouthful.

It's a Vagational Beja Monte Carlos.

So that's it was I did I didn't check any any PR or any it wasn't to to find like a

A good name.

But anyways, uh Buds is it's probably nicer, is Bayesian adaptive direct search.

And uh what is it?

So I kind of came earlier, and I I think it's a it's quite uh widespread now.

and uh it's an optimization method.

So I say so unlike so VBMC is you know me trying to do Bayesian inference properly.

So PyBuds or bass it came earlier, so it was kind of okay, I I'm trying to solve this model 15 problems.

let's treat them as an optimization problem.

So let's say you know just you know maximum likelihood estimation.

So you won want to find the parameter that maximizes you know your the probability of your data density of data the your log likelihood, your likelihood.

And um so anyway, if you strip of everything, it's just an optimization problem.

So you can use any optimizer you want to do maximum likelihood estimation.

You you you put you take your favorite optimizer, you know if you can compute the gradient, you compute the gradients and you know and you you go.

so the problem is that most of the problem they say of the models that we had at the time, still nowadays.

in in the in the sciences.

for example, yeah, they're not nice, you you don't have gradients, etc.

etc.

So I was looking so you enter the realm of uh derivative free optimization in the sense that you need to do optimization without having gradient.

That's hard because then you know given a point you don't know exactly where to go because the gradient helps you you know knowing where the minimum is as it's you know a good a

promising direction is when I say without the gradient you need to figure it out.

You know and that's actually I I had a kind of rabbit home for one year kind of studying optimization, you know uh study all sorts of different methods.

But anyway, so I was very unhappy with existing methods at the time.

I was still kind of still doing computational neuroscience and the model fit thing.

So I said, okay, I need to I need to write a better method.

And which works, right?

Because there were many methods which were published, we look very, very good.

And then I any any method I tried broke on on our problems.

Like, you know, there was nothing works.

And it was very frustrating at the time.

Then I realized how, you know, okay, maybe I shouldn't say it, but I realized how you publishing machine learning works.

And um okay, so one thing is what you say in the paper, one thing, you know, then trying to use the method in practice.

And you know, literally most of the papers I tried like broke on, you know, first impact with a real problem.

So I said, okay, you know, I want to solve this.

And I mean, well, solve it was a is a big world, but saying I want to produce something that you know is usable.

Okay, let's let's say it that way.

And uh yeah, so that's essentially BADS.

So BADS i we we talked about Bayesian optimization before.

So BADS is a hybrid Bayesian optimization method.

And the reason why it's hybrid

Is because if you use Bayesian optimization in a in a vanilla sense, like you know, just you know, kind of out of a textbook, it doesn't work.

So that's the secret.

It doesn't work.

So you need to uh use a lot of other tricks which you know modern packages use kind of you know behind the scenes.

Uh but even so let's say many packages would break on our problems because they're kind of very nasty and the Gaussian processes would break in various ways.

Gaussian processes unfortunately are very nice, but also break all the time.

And so

What's hybrid?

So hybrid effectively, you know, you want a method that so when your surrogate model is good, like you know, your Gaussian process is good, then you trust your Gaussian processes

Gaussian process and you know, and that can allow you really to kind of reach the minimum very fast.

So if you have a good approximation, that can be very powerful and you can really jump through the landscape and find the minimum.

The problem is that many times your surrogate is not good.

And uh because you know your function might be very nasty, you don't have enough points to really represent what's going on, and trusting the surrogate would be a terrible idea

then.

Then essentially you want some sort of a fallback uh methodology to kind of you know to still do optimization in the situation, maybe that get out of that region which is kind of

nasty, and then you know, go back to kind of trying to fit things with uh with your surrogate.

Uh now what I said look like it looks unheuristic, but there is a framework which is called mesh adaptive search.

uh which is you know it's it's a paper, it's a very highly set of paper, the journal of optimization, if not mistaken.

But anyway, so it's so that gives you a framework for how to do this.

So what I did essentially I took the framework, it's a framework so you can kind of play around with different components.

And I kind of you know I kind of built my version, which is again BADS.

And the short answer is that essentially so again it's it's it has limits because of course methods have limits.

So say so for example works up to you know 20 ish dimensions because of this idea that uses discussion processes.

This is you can use it with higher than twenty dimension, but then that kind of that part of building a a surrogate a nice circuit breaks down pretty much all the time.

So you end up only with the kind of the other part, which is the kind of slower and not fast way.

So you get you a a bad method.

But if you're in a situation in which uh your model, you know, sorry, your your function can be modeled in in some parts, not in all of it.

So say so you can have bad parts and you know, and the method gets out of those.

But then you once you're not in a good part.

then you know you can you get this kind of acceleration because now you can uh build uh the threat and then you so it it can be very efficient.

And there's this nice thing which is kind of the you know I think Andrew German mentions it in in his blog so the there's this thing that when a model is good like uh and when you

know near the like near the optimum, near the maximum you know maximum likelihood uh value things are nice you know because I mean if this come actually you know from uh the

Sorry, uh Bernstein von Mises theorem.

in the sense that, you know, things eventually, you know, kind of all converge to a Gaussian.

And so I said so near the the kind of the the maximum like solution, maximum posterior solution, everything kind of looks like you know it's nice and quadratic and Gaussian.

And so I said so.

If you're in a good region, things are nice, and then you you can model them nicely.

If you're farther away, things can be nasty.

But the point is, you know, you don't particularly care because those regions are just

Bad.

So you know, you kind of want to e exit those resources quickly and get to the good ones.

So anyways, so that's that's why in a way why uh bads I think also works very well for optimization of especially for models.

Okay.

Okay, damn.

This is this is also super interesting.

I feel like it it works in in a smaller number of cases, right, than than VBDM.

But if you enter that case

That use case is gonna be extremely helpful to use Pybats.

Is that correct?

Yeah, it's actually more the other way around.

Like Pybats is why is more of universal in the sense that a lot of people in in our field I know I know I not say b I think I say you you you're of course to say you from basic

statistics, so you think everybody does basic statistics and that's what I wish.

But no, that's not it the reality.

The reality in in many scientific fields, at least, you know, in in cognitive science and um especially in neuroscience, people

are not at the level yet of doing Bayesian inference and they they are totally okay doing maximum likelihood estimation for a number of reasons which are also understandable

because the models are very complicated so doing Bayesian inference is could be completely hellish.

So I say so so people are totally happy with getting, you know, half it so let's they don't you know they don't have any pretense of trying to find the full posterior

distribution.

So that's that's what's kind of my attempt with the BBMC.

so so that's why BADs you know is way more used as well because it's essentially in the end it's just an optimization method.

With you with some tricks to make it work, etc.

in you in in harder cases, et cetera.

Uh so yeah, so I mean I would say a a lot of people are using Baz and it is a smaller kind of community that uses VBMC.

Yeah, yeah.

Okay.

Interesting.

Uh that's actually a good point.

and I don't know if you already have a mascot for um Bads, but it obviously needs to be a bunny, because that way it's uh it's a Bats Bunny.

You know, and and I think it's it's a perfect marketing tool, you know, so yes, that's that's good point.

Yeah, we need to we need to get them there.

that's how we'll win the battle.

so okay, so that's BBDM z that's bad.

Uh and and and running a running theme, a running topic in your work, especially lately, is neural processes.

You you said at the beginning you really love them,

What was the unlock for you?

What's so fascinating about them?

Yeah, so let's say so it w it was interesting exactly because I said, you know, I was kind of wary of uh of deep nets.

Um because again, you know, I mean just honestly, just because, you know, in a way I did I I didn't understand them.

I mean, let's say I mean yes, okay, I understand them, but you know, I didn't get them because it looks like okay, you just have a problem, you throw a neural network at it, and

then it kind of solves it, but that's a bit ugly.

And uh and then, you know, I mean of course let's say I I I I I'm

Talking like a physicist, you who likes to see symmetry and you know and um and then you know let's say yeah neural processes to to me at least uh they they spoke to me uh exactly

because it's a they were not completely arbitrary, but it's a there was this idea that first, you know, first of all, the very nice idea of this kind of meta-learning in the

sense that you're training on a data set of data sets.

So it's a so you're kind of learning what the task is by seeing a lot of instances of the task.

And so and each it it's nice because each kind of so.

Each instance is kind of like a micro problem that the network needs to learn to solve.

And then you're just given another micro problem and then another micro problem, and then you know, you kind of and it's kind of learning to possibly generalize, you know, across

different data sets.

And that's kind of I find that's okay, that's interesting.

There's something going on there which is which is interesting.

And uh and then they say I specifically liked this idea that okay, so since again, you are conditioning on this available data and this is a set.

You want your the architecture of the network to to respect this uh the symmetry.

And and this is done, let's say, with the old kind of conditional neural processes or neural processes originally, with a deep set, which was proposed in a previous paper, uh a

deep set.

And uh which essentially takes it's a way to encode uh set based information in a neural network.

Again, it kinda you it's it's quite elegant.

And it tells you, you know, how it mean the original paper tells you how you can do it, what are the constraints that you can have, you know, on your architecture to do this.

And yeah, it's uh I mean I say and it's actually quite simple.

Like you know, the b the basic version is quite simple.

Essentially the idea is that you have a bunch of data points, each data points get embedded separately, and then you get a representation and then you just sum all the

representation.

And since, you know, the sum operation, you know, is commutative, you can swap all the orders and you know, I say the the end up the end representation essentially just, you

know, it doesn't change if you change

the the order of your data because say that just you know you embed them and then you sum them.

And so I found that you know it was it was kind of very, very pleasant.

Then you know you can have uh deeper deep sets, but say for example the original neural processes and many other papers, especially kind of stop at one level in the sense that

you know you kind of get these data points, you embed them, and then you know, and then you just sum the embeddings and you get this representation.

And then you know you have a kind of a readout that kind of makes predictions based on that embedding.

Okay.

And so that's why you're saying that transformers are a natural process architecture.

A natural sorry, a natural neural process architecture.

Like I mean because I I've seen you write that and I'm wondering what that means precisely and what that implies for how we f should think about prediction in context learning,

amortized patient inference and and all the the other topics you work on.

Yeah, yeah, yeah.

That's that's a good point.

So so that's why that's that's for me was the bigger unlock.

When I when I especially after kind of learning about new processes, and then a few years later, again, you know, took me a while.

Uh, you know, everybody was talking about these transformer things.

And, you know, twenty seventeen, of course, uh tension is all you need.

But then, you know, Transformer became very big, uh as we know.

And uh and then yes, so let's say so what I described before.

I mean I use that language, but so it was a language at the time.

It's like you know, you have data sets, you embed them, you have a representation, and then you work with that representation.

Uh nowadays we would call them tokens, and but it's exactly the same thing.

So you have each kind of data point uh is is a token.

you embed it in a representation, which is you know your whatever uh and you know D-dimensional vector.

And

Now you wouldn't sum them because summing is what you know what we were doing before in in a kind of standard neural processes in in modern you know in the transformer architecture,

you allow each token to kind of look at each other using the attention mechanism, which is just you know it's a is a more complex operation.

And the beautiful thing about attention is that attention again is a set operation.

So let's say so there is no ordering in a transformer.

Uh since transformers were proposed for

Language modeling, even originally.

So actually they had to break the permutation invariance or equivariance.

So they have to introduce uh what's you know position embedding, which is something like, okay, this you know is the first position, this is the second position.

So it's something that you need to add to a transformer to make it uh aware of the sequence or that it's a sequence.

But otherwise, you know, a transformer as is the architecture is uh you know permutation, say equivariant slash invariant, depending on what you do.

And so they say, so yeah, so you can just shuffle around your data points and you know, and the result doesn't change.

So I say so it's a very, very natural uh kind of architecture to represent the idea of getting information uh from a data set, a set of data points.

Each one is a token.

These tokens you know talk to each other by attention.

You do it multiple blocks, they keep talking to each other, and then in the end you have a representation.

And now you can use that representation to do things.

And

If you think in Bayesian terms, what's going on here is that your network is conditioning on this data set.

And now whatever so it's kind of learning, you know, it's kind of given this data set, then you know you can take, you know, the for example the last layer or whatever, and

then, you know, do prediction based on this layer.

But say what you're doing is you're conditioning on that information to predict other stuff.

And that again, you know, it's it's that's why I I was saying that yes, so transformers are very natural because say, um

In a way they kind of they supercharge, you know, what you know kind of deep sets we're already doing, but say no, deep sets are a less efficient architecture, you know,

transformers do that, you know, you know this very kind of because essentially all tokens kind of can talk to each other and it's a very powerful representation.

Hmm.

Yeah, yeah.

And such a such a universal one also, so that that makes it extremely powerful.

Um and actually, if I understood correctly, you work

on an amortized conditioning engine.

It's called Ace.

Uh and another paper that you have which is called Alin.

Um let's spend a bit of time on these papers, but they're from last year, so just a bit of time and then afterwards we'll we'll go into what you're actually doing right now and your

your current work.

But I think these two

papers and also the blog post that you have where you just you talk about like you you make the case that you can just predict the optimum.

All this work is related and and that's all from last year.

So if you can go first through through ACE, amethice conditioning engine, to to give to give listeners an idea of of what you've been doing last year and maybe there's something

useful in there in there for them.

Okay.

Yeah.

Yeah.

So like yeah so

Essentially a couple of years ago, maybe a bit yeah, two or three years ago, so we started this kind of this agenda in my group, which is we used to call half jokingly the amortize

everything uh agenda.

So I say we want to amortize everything.

Um now, you know, I I'll kind of make a correction later to this, but it's at the time, you know, that's what we were we were were aiming to do.

And again, you know, we realized that okay, so what we just need to do is, you know, build a big transformer that can condition on things.

And uh essentially the the

2025 paper, this amortized conditioning engine.

The idea there is, okay, so neural processes and similarly, let's say um prior fitted or prior data fitted networks, PFNs, which are also uh transformer neural processes, they

have a I mean architecturally they're essentially the same.

there are kind of minute differences.

Uh

But let's you essentially the entered is kind of this the same world of what we call the transformer probabilistic models.

So the idea is that yes, we have a transformer, you can condition on data points, and all the methods until ours.

Let's say so what they were doing is that they were conditioning on data points, and then you know you have some target points that you want to predict at, then you know they would

give you a prediction as you know, a distribution uh expressed in some way, a Gaussian or something more complex.

But anyway, so you would predict it's a devalue of the function at at these other points.

But we wanted to generalize that because say I don't always only want to predict uh values of things, say or or functions.

So say because say so there was at the same time, you know, there was this other kind of nearby field of amortized inference where you want to predict your data points and you

want to predict uh the model parameters that generated those data points.

That's you know, kind of classical amortized inference.

So you you train your model to give it some data points to predict.

uh the parameters that generated those.

And the kind of the field you know is uh is a subfield of simulation based inference in the sense that you use this data that you generated from these simulators to train your

network to you know predict the parameters from uh from the data.

And then you know of course at runtime you give it another data set and now you you know you predict the parameters.

And for me, not not enough for me, but this is

It's the same thing.

Like, you know, so the whole point is that you want to condition something and predict something else.

So let's say whether that's a parameter and you know, again for and for Bayesian statistics, it doesn't matter.

Like that's just a a a random value that you want to predict.

You know, I don't care whether you know it's the value of a function or of a new data points or or it's a latent variable of your model, like you know, a parameter.

It's just, you know, you just it's a random variable that you want to predict.

And so let's say so the idea was to kind of put everything together uh so that

Uh the model doesn't care whether something you know is a data point, like literally a data point or uh a parameter of your model or anything.

So you want to conditional random variables to predict are the random variables.

That's kind of the mathematical idea.

And that that's pretty much it.

And then you also say, so we b we built it.

And the idea then, you know, I mean, say you need to train the usual way, which you know you have you generate your data, uh, your you're for say the only thing you're doing is

that, you know, again, we now when we train our network, sometimes we give it only data and we ask it to predict.

parameters that generate those data.

Sometimes we ask you to predict other data.

Sometimes we ask you to predict both.

Sometimes we condition only on data.

Sometimes we condition uh on data and parameters.

So it says so we can say, okay, this data were generated by these parameters.

Now you predict these other ones.

And so on and so forth.

So since you kinda you a mix and match of what you predict on.

But again, you know, just you you have this mix of random variables.

But you know again, condition on something, predict something else.

That's that's kind of the core idea.

And then the other core idea is that

A lot of tasks and this this kind of com you you know acts back to to my origins in the sense that you know everything is prediction, everything everything is is you know

inference.

So let's say so a lot of tasks in in machine learning are literally just conditioning on something and predicting something else.

And now not all the tasks, but let's say uh an overwhelming majority of tasks.

So let's say so again, you know, we have a few examples in the paper.

Uh, you know, say image classification is prediction.

You want to predict like a label, which is you know kind of like a a big latent variable, you know, kind of uh coarse latent variable.

You in MNIST, you're predicting this latent variable, you know, let's say you zero, one, two, three, or whatever.

And yeah, that's uh that's just you know conditioning on the image and predicting.

for example, in painting is conditioning on part of an image to predict other parts of the image.

Uh for example, you know, optimization is I condition on the points I've seen so far and I want to predict.

the location slash value of the optimum.

And uh and so on and so forth.

So I say, so you so a lot of tasks is just, you know, uh you just need to kind of formalize in and figure out what what you're conditioning on and what you want to predict.

And that's your task.

So let's say so that that's kind of the idea, which you know it goes beyond, you the say the specific example we make in the paper, but say kind of it it's broader and you I'm I

still believe in it.

So that's just kind of p paper number one.

And um

the the blog post, uh just I'll talk about it briefly.

So the problem with this paper, I'll be honest, let's say it was a bit of a on an issue, let's say, and we put too much stuff in the paper.

Let's say that was actually it's it's kinda like a capital scene in in academia because say people will read the

Read the title, maybe maybe scan the abstract, but say, of course, obviously, but this I le I mean it's in a good way, that says that yeah, yeah, nobody's going to check, you

know, whatever our appendix to see what what else we did.

And so that was uh honestly our mistake.

So I say, so there are too ideas in this paper, too many things.

So I say, so for example, the blog post I wanted to get to highlight some of the stuff which kind of went, you know, a bit uh lost in the message.

And um which is a guy, for example, you our application.

So w as a side kind of minor thing we we did, we know we had this whole section on Bayesian optimization.

How overall, let's say you know, when you do optimization, let's say you did you can think of it as yes, you know, I want to condition some data to predict uh the location/slash

value of the optimum.

And the point is if you if you put it that way, a lot of existing methods that you know, for example, you know, there are methods that uh so sometimes you know you for example you

have some hunch of where the loc the where the optimum might be.

So you might want to put that as a prior over the location of the optimum.

Now, that's

incredibly hard to do using Gaussian processes.

So let's say so so you can define the shape of the of the EGP.

Uh you can tell you know it's smooth, not smooth, etc.

But tell telling a Gaussian process, okay, I really want the minimum to be somewhere here, good luck doing that.

Like so it's it's really hard.

So so there are methods that have been developed, you know, for example, to to do that specifically.

Or for example, you know, suppose you know that the the optimum has value zero.

For example, you you're minimizing some some kind of loss function which is defined mathematically, and you know that you know the optimum is at zero.

So the value of the optimum is zero.

So say so so that that should be giving giving you some information that you can exploit.

Now, again, putting that into a Gaussian process, very hard.

In fact, you know, there are specific papers that solve this specific problem, right?

You know, okay, so how do I tell how my GP that you know my optimum is at zero?

Or how do I modify my Bayesian optimization in such a way that, you know,

the optimum I can't you know sample from this region because I believe the optimum is around this region.

So once you see it probabilistically, yeah it's super easy.

Like you know, yeah, you have a prior on the location of the optimum.

Yeah, okay.

You can just put a prior on location of the optimum.

Or I can condition on my knowledge that the value of the optimum is zero.

Then you know so in A's what you do is you condition on you know y star equals zero, you're conditioning on that and you're predicting and you know.

So anyway, so this we do as a kind of like a a a minimum like a side quest hidden

in the appendix pretty much.

And so I say that should have been a separate paper, obviously.

But let's say, well, that's not how we were working at the time.

So anyway, so that's why I bought I wrote a blog post because I want to kind of kind of just, you know, so everything is about prediction.

And um so we can just of course it's the the the title is a bit of a provocation in the sense that it's not that easy.

So there are some caveats, but let's say but it's an interesting way of thinking about it.

I say so think essentially kind of think of your problem as a prediction problem and you know

How far can you go just thinking of okay, I really want to predict this.

Can I do it?

Um it's not the full story, but so you know you can you can do a lot just just with that.

Hmm.

Okay.

Yeah.

Yeah, and I encourage listener to to check this blog post out.

It's it's really good.

Um you can you can skim it and get the idea pretty fast.

The links will be in the show notes for this ace paper, this blog post, and also the ALINE paper.

Um I do want to make sure you can talk about what you're doing currently.

So um maybe we'll we'll be able to talk about ALINE after once.

But first, you told me while preparing for the show that your group has uh several paper had several papers at ICLR this year.

Uh and you had a bunch of really practical things uh to talk about here and you're doing very interesting things.

So yeah, can you

Walk us through that and and tell us what you've been up to lately and where where your mind is at.

Okay.

Yeah, thanks.

Yeah, so I'll I'll try to be to be concise.

So we have time here to touch about the these different points.

Yeah, maybe there are a couple of papers that you know are kind of more practical uh from our kind of ICLR batch.

One one is this uh it's called a PriorGuide, and the idea is

Suppose that you okay, you resulted pre-training that you can do, let's in amortized inference.

You know, you train your your neural network model uh to do inference.

Then you so you spend it doing this kind of relatively lengthy uh pre-training effectively.

And then you know you deploy it, and then you are at runtime, okay, you give it a data set, and then you know you want it to give it this data set, for example, giving you give

you back the posterior distribution of the parameters that might have generated this data set.

Now, it's a posterior distribution and

As the audience will know, obviously let's say that depends on the prior and the in your data.

So how do I specify the prior?

Uh so okay, so of course let's say you can specify a pre-training, but suppose that you know you're kind of pre-training a model and you know you get some additional information

later, which you know modifies your prior in the sense that, you know, or you know, let's say another situation you can think of, especially in kind of as we go towards foundation

models for inference, let's say someone else pre-trained the model, okay, with a specific prior.

maybe a broad prior, you know, which kind of accounts for many phenomena.

And now you know I want to use that model for something more specific.

So I know that you my my prior is is is more specific.

So if I use that broad prior, essentially I'm kind you know throwing away information I could use.

So how can I inject this information, you know, when I use the model?

Now that's that's a big question.

we try to do that also with ACE to a degree, but say we we kind of tackle you know more aggressively in this kind of more recent paper, a PriorGuide.

And the idea essentially we use the machinery of uh diffusion.

I'm not entering in the in the details, but say you had also an episode with uh Jonas Arruda.

So let's say it's probably it's very related.

So the idea is that so we use the machinery of diffusion to kind of so you can use diffusion again as a method to give your distribution.

So let's say so pip you know, we can use do simulator based inference uh to condition on some data, and then you know you do your kind of diffusion thing, and then let's say you

get a dis samples from your posterior.

Good.

Now the nice thing about diffusion is that you can use uh guidance, which is kind of this technique which uh essentially you kind of steer the the diffusion, which you know the

sampling in one direction or another.

And if you kind of choose your guidance in some you know uh ad hoc way based on your problem, you can effectively you know have your model do different things at runtime.

And the idea is that effectively, you just in a in a nutshell that we find a way we find a way to encode the guidance which kind of moves

the the sampling in such a way that you can use a different prior at runtime.

So say so you train well you or someone else trains the big model uh you know on a broad prior, like you know, because you imagine that you kind of foundation like, so it's very

broad, many different parameters.

Then you know at runtime, uh yeah, I have my experiment and you know I know actually that you know the parameters tend to live in this regional space.

I can put that prior uh at runtime using you know our technique, for example.

And again, and it's part of a broader

kind of area of research that we are really interested in, which is again this idea of yeah, test time compute.

I mean, which of course is very relevant also to LLMs, but say, you know, so essentially how can we add information or you know additional computation at runtime?

Before I was saying, you know, we want to amortize everything.

That was you know a few years ago.

Now I kind of, you know, there is a caveat to that.

Let's say there is an addendum, let's say that yeah, so maybe we cannot amortize everything because that would mean essentially kind of

pre-compress the entire universe in a single model.

Maybe we cannot do that.

There will always be some situation that you know you have not quite encountered before and you need to kind of spend computation at runtime to figure out what's going on.

And uh so I said so yeah, so we're kind you know following the trade, you know, in a way the the the the steps of you know LLMs and agents in the sense that uh yeah, so you do

some pre training, but then you probably also want to spend some time at runtime, because you know you have some information that you get only at runtime and now you

You want to use it.

So PriorGuide is just one specific instance of that.

There might be, you know, well, there are surely many others that we are exploring.

Uh no.

That's what kind of so it's part of a broader agenda.

So how do we do, you know, amortized inference?

But again, not everything is amortized, something might be changed at runtime, et cetera.

So it's it's very exciting because there's other stuff that can be done.

And uh the other paper that I want to mention very briefly is this efficient auto-aggressive you know uh inference for a transformer probistic models.

And actually

Kind of the the the the tagline is is is kind of is very short.

And the idea is that okay, so I I discussed before how it's important that uh all these models uh kind of preserve uh this set property.

Uh you know, so let's say so you can kind of move around your data points and you know, but let's say your the your prediction doesn't depend on the ordering of your data because

it's set.

Now it turns out that that's all very good and nice, but it makes really nasty effectively kind of expanding this set.

So if you have got more data.

since it's a set and it's using kind of this self-attention, which is you know kind of between all the points, every point that you add, you need to recompute everything from

scratch.

So there is no concept of for example, you know, KV caching, which is you know kind of a staple of modern you know transformers and modern LMs.

The idea that you know you kind of store some computation you've done so far and you only need to compute the additional tokens that you learn.

So the idea is that of course we could do that, but we would possibly learn to possibly lose something.

So the idea is that okay, so what if

We we use this idea of a buffer in the sense that uh so we keep the context you know fully you know set-based, fully self-attention.

Then you know we can add kind of something which is doesn't do full attention with the entire set, which would require us to recompute everything, but effectively only does

what's called causal attention, which means that each tokens kind of only watches the previous tokens.

So now this is not permutation invariant anymore.

So we're kind of breaking this permutation invariance.

But we are doing it only for kind of these additional points that you add.

So the idea is that you do kind of add new points very quickly and you can make predictions very quickly for new points.

And then you know, once you're done with that, you can kind of okay, you know, now you have time, you can put them back into your set.

And now you can do you can use low update in which you you update everything.

So kind of you know, so and this actually has a lot of usages because it's a because it's kind of so you can keep your context, you know, permutation invariant.

Then you know you can do a lot of kind of even in parallel, you can literally sample, you thousands of points, you know, using this technique.

Um and then, you know, let's say you can kind of merge it back, all the information back into your context.

Anyways, so it's it's maybe in in a way it's kind of more, yeah, kind of architecture, more engineering in a way.

Let's say there was a lot of problems that need to be solved, like engineering wise.

Uh, but it's a yeah, it's it's something that we're we're kind of building on because say it kind of lets us again, it's an unlock because we can do a lot of things with this now.

Damn, yeah.

Yeah, it's we definitely need the

The links to these papers in the show notes.

in so what what are the the cool things you we can do now with that?

Like I what are you working on right now based on these papers?

I'm guessing you're trying to add some of these um possibilities to to some packages.

So yeah, what uh what are you planning around that to help people use it?

Well

So the the the grand vision here is that we've been discussing say my group, let's say, you know, I I have the you know I can I can show the the logs.

I've been talking about you know foundation model for inference for for many, many, many years.

And and um so that's something you know we've been kind of discussing for a long time and that's always been kind of the you know the north star for us, for my group.

Especially you know, building exactly this foundation model for inference.

And in order to get there, yeah, we kind of you know need to solve a lot of problems.

shout out of course goes to what have become foundation models for inference recently.

I mean recently, past few years.

Uh all the tabular foundation models have been kind of have shown us the way how you can build uh a foundation model for inference.

Uh of course they say they apply to tabular data, but I mean a lot of things.

can be put in a table.

So it's a so from from my perspective, let's say they're kind of there are general purpose foundation models for inference.

Let's say even though people cannot call them tablet data for historical reasons, I guess.

But let's say so you can apply them to a lot of things and you know and they've been kind of doing you know amazing.

Uh so we are very interested essentially in a in a in that direction, to expand in that direction.

And there are a lot of things you can do.

Let's say but you say so of course it's a um there is a lot of

New ideas that can be tried in the space.

And luckily these models are still fairly small, let's say, compared to LLMs.

So they're still nicely within the reach of what we can do again, you know, within academia.

So we don't need to kind of spend millions or even hundreds of millions or billions to get uh training done.

So let's say so so that's quite exciting because we can still kind of contribute.

And for example, yeah, just you know, to mention the buffer that I just mentioned, yeah, can be applied to

Transformer, neural process, etc., but also to tablet foundation models.

So let's say so uh one particular thing that you can do with this kind of fast prediction that doesn't require you to recompute everything, uh, is unrolls, which are very useful

for if you're doing reinforcement learning.

So if you kind of want to do you know fast predictions to see what happens, you know, as you add more data points and you want to do you know RL on this, uh the what we have

essentially would be built uh allows us to make it like

hundred times faster or something like that.

So it's no, we know with with much you know, with a hundred times here fewer memories.

Like it's it's like yeah it's it's super useful.

So let's say so that's something we're going to use definitely.

I mean also others can can use.

And just you know just mention a direction.

So a lot of the things we are doing are part of a kind of a grand plan.

And you know we're kind of building one step at a time.

But let's say there is there is a there is an arc where we're we're going towards.

Yeah, actually you wanna talk about that

Long arc or do you wanna talk a bit more about ALINE, whatever you want, because I I do wanna be mindful of your time and and start playing us out here.

Yeah, yeah, thanks.

So I'll I'll give you maybe I think line ALINE doesn't doesn't take a lot of time and actually it's a good segue to kind of discuss more on the um foundation model direction.

So the idea of ALINE uh it was to uh take what we did with Ace.

So I say you can predict things, you know, you can condition things, you can predict things.

And now we add the fact, okay, we also want a model that does active data acquisition, which means, you know, because ACE is passive, it's just doing Bayesian inference, you

just feed it the data, whatever data you have, you feed it whatever information you have, and then you know that's as predictions.

Now we also want to collect data, you know, do active learning or

Experimental design, which means essentially you need to select which points in the space.

And again, your Bayesian optimization is a form of active learning, right?

You're you're sampling points in the space so that you know you want to find the minimum.

So that's the that's what we want to do.

Uh we had some work before uh with Dao Lang, and this is done for work done um with particularly with Sami Kaski at university of of Aalto University.

And um

So Dao Lang is a PhD student shared between uh Sami and me.

And uh the idea is okay, so let's use effective reinforcement learning to train um another head on top of let's say an ACE-like model that instead of predicting just the value, you

know, say data predictions, it works as a policy head and tells us which points to to collect.

And uh was the what's the goal here?

Well,

The goal here, for example, can be our ability to predict other data points.

And and this is kind of kind of self-reinforcing in the sense that you have one head that you know makes predictions, then the other head that you know kind of computes the policy

that lets you predict better.

So let's say so these two loss of functions in a way kind of work hand in hand.

And uh so we have a single network that does both.

And of course it's con

is both convenient in the sense that okay, so now you have a single network that you know essentially you can kind of use it for data collection and then you can also make

prediction with it.

Nice.

But I say I think it goes kind of deeper than that because say now we have a single network which is learning to this use task.

And these kind of tasks help each other in the sense that the prediction head is actually used as a target for the policy head in that, you know, use the policy head to want to

improve the prediction, to make, you know, for example, you know, more precise, improve the

The put the log density, et cetera.

So again, it fits uh yeah, there's this nice synergy between the heads.

That that's the idea.

Then you this trained via you know enforcement learning and so on and so forth.

So then you there are lots of kind of technical aspects, but it's you know, kind of the the the high level idea is is this one.

And then you know, but this brings me to the kind of foundation model idea.

Because if you look at many, say the modern foundation models, like you know, again, like in this and again in in this field, uh you can think of tabular foundation models because

those are kind of you know

what occupied space of course you there are there are others the point is you know that they do prediction so you just you know fill in more points etc but i say so yeah the

division i have for a foundation model is is more than that it's so of course it's idea is a core of prediction and inference of course you know as we said it's very important it's

you know as the base and the the basis well the base of everything and uh sorry I had to do that but the the point is

We we want to do tasks with it, right?

So, you know, say you want to be able to do for example, you know, again based optimization is an example.

experimental design is an example.

Again, the goal is not similar.

So it's not not the same, but you know, it's similar.

Uh so you want to collect data so that you know, uh you want to improve your predictions for, you know, for certain certain parts of your space, for the parameters.

Uh you know, so you might have a task.

So say so and the point to me let's say that that's these these goals are not kind of, you know, an appendix.

the model, but say to to in the I see that as part of your foundation model.

So your foundation model say you know has kind of an array of tasks that is trained to do.

And the point is, I mean, again, uh the and this is where I think let's say the we we will benefit from the magic of of meta learning in the sense that the the the model, you know,

again if it's big enough, is it's learning all these kind of representations to do all of these tasks.

And the point now is kind of using them, right?

You know, so so it's learning to to generalize between different or at least interpolate

Between different tasks.

And then of course it's a yeah, I kind of spilled the beans in the sense that so of course you you hope that you know then at some point you start seeing some degree of

generalization.

And you know, so essentially as you do more and more tasks, not just prediction but other tasks, let's say, you know, uh yeah, the you you you get something more uh and again, you

we we have glimpses of that with you know, with again with agents LMs, let's say clearly, of course, yes, you know, they again it's very jaggedy.

As we know, et cetera.

But still it's a yeah, I mean, they're definitely not just literally doing what they're training on, but say there's a bit of uh, you know, jagged generalization and maybe

limited, but still, you know.

I mean, I'm I'm pretty sure especially a lot of the problems that I use them for, you they've not been trained exactly on that.

So anyway, so let's say so again, you know, can we can we do something similar?

Would that even you know, what would that even look like?

Like you know, so it's it's not obvious, right?

so I said there are many, many, many open questions here, and I think let's say, but that's anyways, that's what I see as the big direction.

So eventually again, the dream is this kind of big oracle foundation model, which you know, you you know, you can and you know, and LLMs might be a part of it.

Uh I know that you asked, you know, I discussed it with with this Stefan.

And um I think maybe we have a slightly different uh version of that.

Of course they say yes, uh I I totally agree with you know, uh agents that, you know, like modern agents that just use kind of Bayesian tools as, you know.

say as tools and that's that's definitely let's say I I see that totally happening.

I also say I I think we can also push on the concept of big models for inference that just you know that just are focused on doing inference and learning to do inference and you

know and they're good at doing inference.

And then on maybe these models can be then called by agents okay.

So the the two things are not um kind of opposed.

Yeah, yeah exactly.

Uh I think we

We we spoke about that at length with uh Stefan during two episodes, so definitely recommend listeners.

Um listen to these ones.

The one with Jonas Arruda also was a bit more focused on diffusion models and he also um did a live a live demo also.

Um definitely recommend that one too.

And yeah, in general, really agree with with what you just said here and

I think it's a very important and interesting part of your research, not only yours, but like your your whole group where you're that's really interesting to see how the NLMs and

agent part or merging with the amortized inference part and the and the more Bayesian models side of things.

Um and actually of course you've been working a bit about that right now.

You're also

Doing some work on L LMs and agents.

So yeah, what I'm I'm curious what you're mainly doing right now about that and and what was the question, you know, that pulled you in and how do you see it connecting back to

amortization and probabilistic conditioning?

Yeah, so the the w so the work strictly on LMs and and agents, let's say still in stealth, I would say, in the sense that we haven't published.

Uh so interesting.

So we are yeah, we are we're working and that's let's say that's that's ongoing.

And um although let's say so we it's an idea that you know some some of these ideas have been kind of going back and forth.

but let's say so the angle that we were exploring at the moment is related to there was recently a a position paper.

Uh this idea say the idea I've been working on, let's say we've been working on it say come earlier, but of let's say you it's in the the zeitgeist.

And um so let's say so there was this kind of large position paper recently that says, you know, we should think of agents as, you know.

uh In the context of uh Bayesian decision theory.

And uh and more than ever, effectively we need Bayesian uh decision theory.

Uh because you know essentially that's what's happening, it's not making kind of small decisions all the time, constantly.

And so I say so that that's kind of my my the hook for me was kind of start to play around with um local LMs, small agents, um, because they say I'm interested in resource

constraints.

We haven't talked about that at all, but it's a you know one line of research in computational neuroscience has to do with essentially inference under resource

constraints, and we have some work on that in more on the neuroscience camp, binaries.

But it's so I think that's very interesting because say that there is this idea that intelligence kind of you know arises under uh boundedness of resources.

So let's say so you need to, you know, the idea that so

The fact that need to compress memories actually makes us kind of generalize because we need to kind of compress things to then generalize.

Anyways, so I think the whole idea of yeah constrained uh resource constraint system is very interesting in itself for for a whole lot of reasons.

And so for these reasons, the local LLMs or you small LLMs or tiny LLMs even are very interesting.

But so you want to them to be able to do things, um but of course, you know, they're very limited.

So I say, so how do you make them more usable, more reliable?

And well, it turns out that you know I have a lot of work on sample efficiency, et cetera.

So essentially, so I've been kind of combining these these ideas from sample efficiency uh in the kind of small, I mean LM world, but say of course the application mostly would be

for small LMs.

Also because they're local, you can play around with a lot of stuff with uh the way in which you know you sample tokens from from this.

Uh and you know to to be concrete, for example, you know, if you think about uh what uh small LMS do or small agents do, for example, you know, when a small LM uh in general,

also agents in general, you know, decide to call a tool effectively when it kind of needs to emit a token, you know, uh okay, I'm I'm calling a tool, and you know there's a

specific token for that that changes dep depending on the on the on the family.

And you know, and then you know it says, so okay, now I'm calling a tool or I'm not calling a tool.

And you know, if I'm calling a tool, okay, you know, then you if I tell you you for example, tool name, uh, you know, it's it's another kind of token, or maybe a few tokens.

And you c you can think of that or you know, as kind of small choices or small decisions that are made.

And they're probabilistic in the sense that, well, first of all they they they're well, they're stochastic because LMs tends to be stochastic, you know, unless you put the

temperature to zero.

Also they're probabilistic in the sense that you they they based they're based on limited information.

So, you know, you can think of, you know, effectively you can think you can build a posterior over this and, you know, and work with that and do kind of the old Bayesian uh

stuff.

And so anyway, so it's that's kind of the the things that uh we've been exploring.

Okay.

Very, very interesting, very cool.

Uh definitely sounds like uh worthy of another episode when you guys have that out.

Um yeah, so as listeners probably know I've been also working myself on some

And some Asian stuff.

I'll refer you to at least episode 157 and 158 with Stefan.

But also you can check out the blog post section of the website because I've also worked on a Bayesian workflow scale and a causal inference scale.

The one with Stefan was really about the the amortized workflow scale.

And yeah, um, I mean that's also why I I love talking about you guys too on the show because it's also something I'm I work

on myself and I think has a lot of potential to um help people in the future and make sure the good practices that come from latest research actually uh gets distilled into the

general public.

Uh so so yeah thanks a lot for for doing that Luigi.

And actually I'm also curious, you know, to finish on a practical note for an LBS listener who

currently lives in Stan or PyMC or an NumPyro, where should they actually start with um the tools we've mentioned today?

So mainly PyVBMC, PyBADS, BayesFlow.

What's the smallest interesting problem to try first, you would say?

Yeah, that's that's interesting.

So

Yeah, these essentially yeah the tools we we we discussed and then uh yeah base flow essentially which you know which kind of like a general purpose amortized inference um

tool.

Kind of apply it to uh mildly different situations.

And we've been trying to kind of combine them in the sense that, you know, with with our uh amortized Bayesian workflow paper, again with with Stefan uh and others and it was led

one yeah by by my now graduated student Chen Kun Li.

And um

So ideally you want to start actually with amortized methods because say they're especially if you have many data sets to to analyze, you know, that that you can have a a

very strong, you know, win there.

Uh because say, you know, if you have many data sets to analyze for the same model, then you would need to r run, you know, MCMC or, you know, any method from scratch and that's

that can be very expensive.

So let's say so amortization would pay off.

so let's say so you if you in that case, let's say you should consider using amortization.

And then you know, for example, I mean, of course it's a bit of a plug-in, but uh we have these uh these these ideas for how essentially you know you can then check whether your

amortized methods um you know, say your amortization has has worked and or you know you can trust it.

And the point is of course if you don't trust it, then you know, for those specific data set that you don't trust, your those posteriors that you don't trust, then you know you

can do kind of MCMC and you know in a way which by the way is supported by amortization.

So you can use amortization in in some way.

So that's that's essentially kind of a a a natural way for for example someone who does T C and does not trust amortized methods.

And fair enough, you know, let's say I I I I understand, you know, where this kind of um mistrust can come from in the sense that of course C C has built a lot of diagnostics over

the years.

And you know, it took many years to come, you know, to the modern diagnostics, right?

So let's say so that's

so you know that's something you know we we we in the community uh are are are building also for amortized inference, right?

So say so because again, uh understandably, in order to be use a method, you need to know when the method failed.

It's probably one of the most important things uh to to know uh with a method uh as a practitioner.

And um so I say so yeah, the so essentially, yeah, many data sets, then you know you should definitely consider amortization, uh, because that might really kind of speed up

your workflow and you know let's say yeah if you look at our our tricks, our suggestions, then you know you can still you know kind of trust the the whole process because in worst

case scenario you kind of go back to MCMC for those specific problematic data sets.

That's one thing.

Then you know there is the other angle which is if your model is as we discussed, let's say expensive in the sense that you know each evaluation of your uh log density of your

model

takes time, or we know has a cost of some in some sense, you it typically the cost is time in computation.

Then you know, then is you're in the case in which you might want to look at for example, PyVBMC.

And to be honest with the I I'd love to connect uh PyVBMC more to the PyMC and uh base flow environments so that you know they could kind of speak more to each other.

And that's something that you know we've been discuss discussing but say yeah we haven't managed to do yet.

But now you know with the agents, you know

Maybe we can kind of speed that up.

And so anyway, so if you have something expensive, etc., then you know definitely let's say uh you might want to consider methods that kind of do efficient Bayesian infest in the

sense that they can get you to an answer uh with a relatively small number of evaluations of the log density, like in again of the order of you know, maybe up to you know a few

hundreds.

And that that can be you know, of course, uh kind of very useful.

And finally I would say, you know, uh but of course, you know, that's not something necessarily I would say for

in a in a Bayesian statistic podcast, let's say, you know, so there are people who don't care about no say for that number of reasons, let's say they might be okay with a point

estimate.

And of course there are reasons to find a point estimate because for example, you know, um even if you're doing uh you know if you want the full posterior, uh, you know, doing like

a a a rough optimization, maybe you know, you don't want to start you know exactly from the map for a number of reasons, especially if you're in you know moderately high

dimension.

But if you're in a relatively low dimension, you know

There's nothing particularly bad in in starting, you know, near the map, like the the maximum a posteriori or or mode of your posterior.

Uh for so and in typical, you know, actually, you know, it's it's a decent starting point.

Um again in low dimension.

In high dimension things are very different, so you shouldn't do it.

But let's say in in in low dimension, let's say so again, you know, how do you start from the mode?

I mean, you can run optimization.

And in fact, let's say that's for example, how many practitioners use kind of pie buds in combination with Pi VMC.

So first

They run optimization such that they would kind of find you know a place uh near the optimum, which would be the mode, and then start PyVMC from the mode.

And that actually works well because as we discussed before, you know, typically the posterior is kind of nice near the mode and might be quite nasty elsewhere.

So I say so if you were to start PyGBMC or even MCMC from you know from farther away in details of your posterior.

Things, you know, can be quite nasty and bad, you know, and you'll say have kind of numerical all sorts of numerical issues.

But you know, if you kind of start near the mod, things are nice.

So anyway, so say that's how kind of you you can kind of this kind of this nice synergy again within PyButs um PyMC.

Yes, definitely.

Uh and I know also some work has been uh done lately, especially by the by uh Alex Fengler, who'll be soon on the show.

I'm working on that.

making a bridge between

Time C base flow.

And so yeah, like definitely that resonate with uh what you've you've just mentioned.

Uh fantastic.

Luigi, any any other topics you you would like to discuss?

One thing I say, but you know, I want to kind of state again is that uh the the motivation here is always practical in the sense that we want our tools to be used, right?

So we are not doing I mean especially you know the trader aspects is very interesting and but let's say so we want to

build tools.

So say yeah, so if you you know if you kind of need to categorize what we do in our lab, we definitely, you know, we are we are tool builders.

Uh and you know we build even complicated tools, but let's say but the idea is that eventually we want people to to use what we're doing.

In the case of for example, you know, some of our methods say it's kind of taking a bit longer because let's say we are kind of building towards the the the releases in a way of

you know things that we can actually commend.

ah But yeah we we we'll you know I I'm very confident you know we will get there.

But say for the feedback from you know users, et cetera, it's actually super important for us.

Like, you what are the interesting problems?

Anyways, also, you know, if if if a method fails, let's say we are actually very happy to hear and yeah, I've been in touch, you know, with many users, like, you know, to know uh

what works and what what doesn't.

And you know, and it's it's kind of keeps research very healthy because it's a okay, these are the problems, you know, uh that you know people care about.

Yeah.

Yeah, definitely resonate with that as

As everybody knows, that's also why why this show is here.

So thank you so much, Luigi.

That was that was really great.

Um and same, I could I could do a a six hour uh ep stop with you.

But I feel that would be much better to do it in person.

So let's uh let's postpone that till I look forward to that I visit House and Gay.

Yeah.

in in the meantime, I'm gonna ask you the last two questions.

I asked every guest at the end of the show.

So first one.

If you had unlimited time and resources, which problem Sorry, uh problem with my with my mic?

I'm telling you to the keys a bit uh Okay, so let's do that again.

Um in the meantime, I'm gonna ask you the last two questions I ask every guest at the end of the show.

First one, if you had unlimited time and resources, which problem would you try to solve?

Yeah, I had to uh well

I mean, okay, so I can give you the kind of gut answer, which is if you look at the the title of you know well the name of my of my group is called, you know, the the machine and

human intelligence.

And it's not just a a fancy name in the sense that, you know, the underlying kind of you know, the fill rouge, the connection uh is that I'm interested overall in yeah, how

intelligence works and and not necessarily the human intelligence, that's the point, right?

Say so and even an animal.

So let's say so for it's kind of all the same in the sense that it's some sort of uh general uh concept underlying, you know, say how how do intelligent agents work, how does

that arise?

And of course let's say the the the work I've been doing, let's say it's kind of very takes a specific angle, which is intelligence has to do with inference.

It's uh which of course is kind of it's a very opinionated take on what intelligence is.

And of course say it doesn't end there, but it's a you know there there is a you ca you can say it covers a lot.

And some people say it's actually everything there is.

there are you know active inference camp, et cetera.

But anyways, that's all another discussion.

And so anyway, so yeah, so big question is okay, what what's intelligence?

How does intelligence arise?

So what makes you know a brain, you know, work intelligently and you know anal like an agent, you know, and then people yes, people are you know asking, are you know modern as

of twenty twenty six agents?

um intelligent.

Some people will, you know, yeah, scream and say, no, that's not intelligence.

Uh that's, you know, whatever.

And that's, you know, just talking predictions, which you know.

And other people will say, Yeah, of course, you know, I I see some elements of intelligence, et cetera.

So anyways, so understanding what's what's the secret of of intelligence, I think they said that's you that's one of the most interesting questions.

Hmm.

Yeah.

Yeah, definitely agree.

that would and that would uh solve a lot of problem

of problems also, you know, as as a trickle down effect.

So uh so definitely agree this is this is a good one.

Uh because it's not it's not a contained solution.

So very interesting.

Um and second question, if you could have dinner with any great scientific mind, dead, alive or fictional, who would it be?

Yeah, that was a that's a hard one, of course.

Um so again I I didn't give it too much thought because otherwise kind of cheating.

And

I would say uh Johnny von Neumann and as uh most of the rest essentially yeah I'll be interested kinda you know uh Piccoli's brain, which is of course famously very uh very big

and uh a smart person and like you know what what is the things about yeah the modern you know developments you know in artificial intelligence, etc.

I think you'd be very fascinated by it.

So I yeah, I don't claim I would be able to handle

much at any in any way is interact, but I would definitely kinda benefit from, you know, kinda okay, uh, you know, given the insight that we have just by being born in in this

area and having seen this explosion of artificial intelligence, uh okay, so what what what does you know, what would he make of it?

And that I would be very interested in that.

Yeah.

Yeah.

Uh for sure.

And and for sure a dinner where there is a lot of things I would not understand.

So

Let me know, let me know and I'll I'll come up.

I will for sure learn a lot.

Um awesome.

So please make sure to add the the latest papers you've mentioned uh in the show notes because I have the ones from last year, but not the ones you've been working on this year.

So uh add that to the show notes.

I'm sure people will will look into that.

So yes listeners, make sure to look at the show notes if you wanna dig deeper.

And WeG, thank you again for

Taking the time and being on this show.

oh

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