Folks, strap in, because today's episode

is a deep dive into the fascinating world

2

of large -scale machine learning.

3

And who better to guide us through this

journey than Sam Dofeld.

4

Currently honing his expertise at normal

computing, Sam has an impressive

5

background that bridges the theoretical

and practical realms of Bayesian

6

statistics, from quantum computation to

the cutting edge of AI technology.

7

In our discussion, Sam breaks down complex

topics such as the posterior's Python

8

package, minimatch methods, approximate

inference, and the intriguing world

9

of thermodynamic hardware for statistics.

10

Yeah, I didn't know what that was either.

11

We delve into how these advanced methods

like stochastic gradient MCMC and Laplace

12

approximation are not just theoretical

concepts but pivotal in shaping enterprise

13

AI models today.

14

And Sam is not just about algorithms and

models, he is a sports enthusiast who

15

loves football, tennis and squash.

16

and he recently returned from an awe

-inspiring trip to the Faroe Islands.

17

So join us as we explore the future of AI

with Bayesian methods.

18

This is Learning Bayesian Statistics,

19

Welcome to Learning Bayesian Statistics, a

podcast about Bayesian inference, the

20

methods, the projects, and the people who

make it possible.

21

I'm your host, Alex Andorra.

22

You can follow me on Twitter at alex

-underscore -andorra.

23

like the country.

24

For any info about the show, learnbasedats

.com is Laplace to be.

25

Show notes, becoming a corporate sponsor,

unlocking Beijing Merch, supporting the

26

show on Patreon, everything is in there.

27

That's learnbasedats .com.

28

If you're interested in one -on -one

mentorship, online courses, or statistical

29

consulting, feel free to reach out and

book a call at topmate .io slash alex

30

underscore and dora.

31

See you around, folks, and best Beijing

wishes to you all.

32

I'm Sam Duffield, welcome to Learning

Bayesian Statistics.

33

Thanks, thank you very much.

34

Yeah, thank you so much for taking the

time.

35

I invited you on the show because I saw

what you guys at normal computing were

36

doing, especially with the Posteriors

Python package.

37

And I am personally always learning new

stuff.

38

Right now I'm learning a lot about sports

analytics, because that's a

39

Like that's always been a personal pet

peeves of mine and Bayesian says extremely

40

useful in that field.

41

But I'm also in conjunction working a lot

about LLMs and the interaction with the

42

Bayesian framework.

43

I've been working much more on the base

flow package, which we've talked about

44

with Marvin Schmidt in episode 107.

45

So.

46

Yeah, working on developing a PIMC bridge

to base flow so that you can write your

47

model in PIMC and then like using

amortized patient inference for your PIMC

48

models.

49

It's still like way, way down the road.

50

I need to learn about all that stuff, but

that's really fascinating.

51

I love that.

52

And so of course, when I saw what you were

doing with Posterior, I was like, that

53

sounds...

54

Awesome.

55

I want to learn more about that.

56

So I'm going to ask you a lot of

questions, a lot of things I don't know.

57

So that's great.

58

But first, can you tell us, give us a

brief overview of your research interests

59

and how Bayesian methods play a role in

your work?

60

Yeah, no, I know.

61

Thanks again for the invite.

62

I think, yeah, sports analytics, Bayesian

statistics, language models, I think we

63

have a lot to talk about.

64

should be fun.

65

Bayesian methods in my work, yes, so at

normal we have a lot of problems where we

66

think that Bayes is the right answer if

you could compute it exactly.

67

So what we're trying to do is trying to

look at different approximations and

68

different, like how they scale in

different methods and different settings.

69

and how we can get as close to the exact

phase or the exact sort of integral and

70

updating under uncertainty that can

provide us with some of those benefits.

71

Yeah.

72

OK.

73

Yeah.

74

That's interesting.

75

I, of course, agree.

76

Of course.

77

Can you, like, actually, do you remember

when you were first introduced to patient

78

inference?

79

Because you had a

80

an extensive background you've studied a

lot.

81

When in that, in those studies, were you

introduced to the Bayesian framework?

82

And also, how did you end up working on

what you're working on nowadays?

83

Yeah, okay.

84

I'll try not to rant too long about this.

85

But yeah, so I guess I, yeah, mathematics,

undergraduate at Imperial.

86

So I think that's

87

I was very young at this stage, we were

very young in our undergraduates, so not

88

really sure what we want to do.

89

At some point, it came to me that

statistics within the field of mathematics

90

is kind of like where I can like, that

should be working on like, applied

91

problems and how what where the sort of

field is going.

92

And that's what got me excited.

93

Statistics at undergraduate are different,

different places, but you get thrown a lot

94

of different

95

I mean, probably in all courses, you get

different, you get different point of view

96

and you get like, yeah, you get your

frequencies, your hypothesis testing, and

97

then you have your Bayesian method as

well.

98

And that is just the Bayesian approach

really sort of settled with me as being

99

more natural in terms of you just write

down the equation and the Bayes Bayes

100

Bayes theorem handles you write down, you

have your forward model and your prior and

101

then Bayes theorem handles everything

else.

102

So you're kind of writing down it's like,

103

mathematicians is kind of like one of the

lecturers in my first year said, yeah,

104

mathematicians are lazy.

105

You want to they want to do as little as

possible.

106

So base theorem is kind of nice there

because you just write down your your

107

likelihood you write down your prior and

then base theorem handles the rest.

108

So you have to do like the minimum

possible work you have your data

109

likelihood prior and then done.

110

So that was that was really compelling to

me.

111

And that led me to a to my PhD, which was

in the engineering department in

112

Cambridge.

113

So that was like, yeah, I had a few

114

thoughts on what to do for my PhD.

115

There was some more theoretical stuff and

I wanted to get into some problems, get

116

into the weeds a bit.

117

So yeah, engineering department of

Cambridge working on Bayesian statistics,

118

state space models and in a state space

model sequential Monte Carlo.

119

And I think, yeah, I mean, for terminology

wise, I use state space model and hidden

120

Markov model as the same thing.

121

So yeah, you have this time series style

data and that was working on that sort of

122

data gave me

123

I feel like this propagation of

uncertainty really shines there because

124

you need to take into account your

uncertainty from the previous experiments,

125

say, when you update for your new ones.

126

That was really compelling for me.

127

That was, I guess, my route into Bayesian

statistics.

128

Yeah, okay.

129

Actually, here I could ask you a lot of

questions, but...

130

those time series models.

131

I'm always fascinated by time series

models.

132

I don't know, I love them for some reason.

133

I find there is a kind of magic in the

ability of a model to take time

134

dependencies into account.

135

I love using Gaussian processes for that.

136

So I could definitely go down that rabbit

hole, but I'm afraid then I won't have

137

enough time for you to talk about post

-series.

138

Let me just say one minute about it.

139

So I'll just say like, yeah, in terms of

yeah, Gaussian process is really cool.

140

Like Gaussian process, like can think of

as like a continuous time or continuous

141

space or whatever that the time variant

access, we'll call it time continuous time

142

varying version of a state space model and

state space model or hidden Markov model.

143

Kind of like that to me is like the

canonical extension of a just a static

144

based inference model to a

145

the time varying setting because you can

and they kind of unify each other because

146

you can write smoothing in a state space

model as one big static Bayesian inference

147

problem and then you can write static

Bayesian inference problems they're just p

148

of y given x or p of yeah recovering x

from from y as as the first step as a

149

single step of state space model so the

techniques that you build just overlap and

150

you can yeah at least conceptually on the

mathematical level when you actually get

151

into the approximations and the

commutation

152

there are different things to consider,

different axes of scalability considered,

153

but conceptually, I really like that.

154

I probably ranted for a bit more than a

minute there, so I apologize.

155

No, no, that's fine.

156

I love that.

157

Yeah.

158

I have much more knowledge and experience

on GPs, but I'm definitely super curious

159

to also apply these state space models and

so on.

160

So definitely going to read the...

161

the paper you sent me about skill rating

of football players where you're using, if

162

I understand correctly, some state space

models.

163

That's going to be two birds with one

stone.

164

So thanks a lot for writing that.

165

The whole point of that paper is to say

that rating systems, ELO, TrueSkill are

166

and should be reframed as state space

models.

167

And then you just have your full Bayesian

understanding of it.

168

Yeah, yeah.

169

I mean, for sure.

170

I'm working myself also on the

171

project like that on football data.

172

And yeah, the first thing I was doing is

like, okay, I'm gonna write the simple

173

model.

174

But then as soon as I have that down, I'm

gonna add a GP to that.

175

It's like, I have to take these

nonlinearities into account.

176

So yeah, I'm like, super excited about

that.

177

So thanks a lot for giving me some weekend

readings.

178

So actually now let's go into your

posteriors package because I have so many

179

questions about that.

180

So could you give us an overview of the

package, what motivated this development

181

and also putting it in the context of

large scale AI models?

182

Yeah, so as I said, we at normal think the

base is the right answer.

183

So we want to get, we want to, but yeah,

we're interested in

184

large scale enterprise AI models.

185

So we need to be able to scale these to

big, big models, big, big parameter sizes

186

and big data at the same time.

187

So this is what Posterior's Python package

built on PyTorch really hopes to bring.

188

It's built with sort of flexibility and

research in mind.

189

So really we want to try out different

methods and try out for different data

190

sets and different goals.

191

what's going to be the best approach for

us.

192

That's the motivation of the Posteriors

package.

193

When would people use it?

194

For instance, for which use cases would I

use Posteriors?

195

There's a lot of just genuinely fantastic

Bayesian software out there.

196

But most of it has focused on the full

batch setting, as is classically the case

197

with Metropolis Hastings, except for

Jekste.

198

And we feel like we're moving or we have

already moved into the mini batch era, the

199

big data era.

200

So posterior is mini batch first.

201

So if you have a lot of data, even if you

have a small model, and you have a lot of

202

data, and you want to try posterior

sampling with mini batches, you want to

203

see how that...

204

If that can speed up your inference rather

than doing full batch on every step, then

205

Posterior is the place for that, even with

small models.

206

So you can just write down your model in

Pyro, in PyTorch, and then use Posterior

207

to do that.

208

But then that's like moving from like

classical Bayesian statistics into like

209

the mini batch one.

210

But then there are also benefits of

Bayesian

211

very crude approximations to the Bayesian

posterior in these really high scale large

212

scale models.

213

So like, yeah, like language models, big

neural networks, these aren't going to get

214

you you're not you're not going to be able

to do your convergence checks and these

215

sort of things in those models, but you

might still be able to get some advantages

216

out of distribution detection, as a

distributed improved attribution

217

performance sort of continual learning,

and these are the sort of things we're

218

investigating is if like,

219

the sort of what if you just trained with

grading essentially, you wouldn't

220

necessarily get these things.

221

But even very crude, crude Bayesian

approximations will hopefully provide

222

these benefits.

223

I think I will talk about this more later.

224

I think.

225

Yeah, okay.

226

So basically, what what I understand is

that you can use Posters for basically any

227

model.

228

So I mean, we're still young.

229

And it doesn't have like the

230

very young and it doesn't have like the

support of, I don't know, if you want to

231

do Gaussian processes, we were not going

to have a whole suite of kernels that

232

you're going to be able to just type up.

233

But fundamentally, it takes any, it just

takes a function, a log posterior

234

function, and then you will be able to try

out different methods.

235

But as I said, like the big data regime is

much less researched, and as much and the

236

sort of big parameter regime is much

harder.

237

at least.

238

So it's going to be, it's not going to be

like a silver bullet.

239

You're going to have to, there's research,

basically, posterior is a tool for

240

research a lot of the time where you're

going to research what inference methods

241

you can use, where they fail, and

hopefully where they succeed as well.

242

Okay.

243

Okay.

244

I see.

245

And so to make sure listeners understand,

well, you can do both in posers, right?

246

You can write your model in posterior.

247

and then sample from it?

248

Or is that only model definition or is

that only model sampling?

249

So it only does approximate posterior

sampling.

250

So you write down the log posterior,

you're given some data and you write down

251

the log posterior.

252

Or the joint, you could say.

253

It doesn't have the sophisticated support

of Stan or IMC or where you actually have

254

the, you can write down the model.

255

but it has the support for all the

distributions and doing forward samples.

256

It leans on other tools like Pyro or

PyTorch itself for that in no other case.

257

It is about approximate inference in the

posterior space, in the sample space.

258

So you can do Laplace approximation with

these things and compare them.

259

And importantly, it's mini -batch first.

260

So every method only expects to receive

batch by batch.

261

So you can support the large data regime.

262

Okay, so I think there are a bunch of

terms we need to define here for

263

listeners.

264

Okay, yeah, sorry about that.

265

Can you define minibatch?

266

Can you define approximate inference and

in particular, Laplace approximation?

267

Okay, so minibatch is the important one,

of course.

268

Yeah, so normally in traditional Bayesian

statistics, if you're running random walk

269

-through troblos -Hastings or HMC, you

will be seeing your whole dataset, all end

270

data points at every step of the

iteration.

271

And there's beautiful theory about that.

272

But a lot of the time in machine learning,

you have a billion data points.

273

Or if you're doing a foundation model,

it's like all of Wikipedia, it's billions

274

of data points or something like that.

275

And there's just no way that every time

you do a gradient step, you just can't sum

276

over a billion data points.

277

So you take 10 of them, you do this

unbiased approximation.

278

And this doesn't propagate through the

exponential, which you need.

279

for the metropolis hastening step.

280

So it rules out a lot of traditional

Bayesian methods, but there's still been

281

research on this.

282

So this is the we saw a scalable Bayesian

learning is what we talked about with

283

posterior.

284

So we're investigating mini batch methods.

285

So yeah, methods that only use a small

amount of the data, as is very common in

286

so it's like gradient descent, stochastic

gradient descent and optimization terms.

287

So hopefully

288

Mini -batches, okay, you said approximate

inference.

289

So approximate, okay, yeah, inference is a

very loaded term.

290

Maybe I should try not to use it, but when

I say approximate inference, I mean

291

approximate Bayesian inference.

292

So you can write down mathematically the

posterior distribution, P of theta given

293

y, and then yeah, proportional to P of

theta, P of y given theta.

294

But that's

295

You only have access to pointwise

evaluations of that and potentially even

296

only mini -batch pointwise evaluation

sets.

297

So approximate inference is forming some

approximation to that posterior

298

distribution, whether that's a Gaussian

approximation or through Monte Carlo

299

samples.

300

So yeah, just like an ensemble of points

and approximate inference.

301

So that's approximate inference.

302

And yeah, you have different fidelities of

this posterior approximation.

303

Last one, Laplace approximation.

304

Laplace approximation is the simplest

305

arguably the simplest in like machine

learning setting approximation to the

306

posterior distribution.

307

So it's just a Gaussian distribution.

308

So all you need to define is a mean and

covariance.

309

You define the mean by doing an

optimization procedure on your log

310

posterior or just log likelihood.

311

And that will give you a point that will

give you your mean.

312

And then

313

And then you take okay, it gets quite in

the weeds the Laplace approximation, but

314

ideally you you then do a Taylor expansion

across them.

315

Second order Taylor expansion will give

you Hessian.

316

We would recommend the Hessian being the

co your approximate covariance.

317

But there are tiny quantities there and

use the Fisher information as said.

318

And yeah, you can read that there's lots

of I'm sure you've had people on the on

319

the podcast explain it better than me.

320

Yeah.

321

For Laplace, no.

322

Actually, so that's why I asked you to

define it.

323

I'm happy to go down into the weeds if you

want.

324

Yeah, if you think that's useful.

325

Otherwise, we can definitely do also an

episode with someone you'd recommend to

326

talk about Laplace approximation.

327

Something I'd like to communicate to

listeners is for them to understand.

328

Yeah, we say approximation, but at the

same time, MCMC is an approximation

329

itself.

330

So that can be a bit confusing.

331

Can you talk about the fact, like, about

why these kind of methods, like Laplace

332

approximation, I think VI, variational

inference, would fall also into this

333

bucket.

334

Why are those methods called

approximations?

335

in contrast to MCMC?

336

What's the main difference here?

337

I honestly I would say MCMC is also an

approximation in the same terminology but

338

yeah the difference is that we talk about

bias and asymptotically some methods

339

asymptotically unbiased which MCMC is

stochastic gradient MCMC which is what

340

Prosterus is as well in some

341

under some caveats, and there are caveats

for MCMC, normal MCMC as well.

342

But yeah, so you have your Gaussian

approximations from variational inference

343

and the applies approximation.

344

And these are very much approximations in

the sense there's no axes on which you can

345

increase if you increase it to infinity or

change the posterior.

346

You cannot do that with the Gaussian

approximations unless your posterior is

347

you're known to be Gaussian, in which case

is more and more I mean, the amount of

348

interesting cases like that like Gaussian

processes and things.

349

But yeah, so they don't have this

asymptotically unbiased feature that MTMC

350

does or important sampling as sequential

Monte Carlo does, which is very useful

351

because it allows you to trade compute for

accuracy, which you can't do with a

352

Laplace approximation or VI beyond

extending, like going from diagonal

353

covariance to a full covariance or things

like that.

354

And this is very useful in the case that

you have extra compute available.

355

So I'm a big fan of the

356

asymptotic unbiased property because it

means that you can increase your compute

357

and safety.

358

Yeah.

359

Yeah.

360

Great explanation.

361

Thanks a lot.

362

And so yeah, but so as you were saying,

there is not these asymptotic unbiasedness

363

from these approximations, but at the same

time, that means they can be way faster.

364

So it's like if you're in the right use

case in the right, in the right

365

Yeah, in the right use case, then that

really makes sense to use them.

366

But you have to be careful about the

conditions where the approximation falls

367

down.

368

Can you maybe dive a bit deeper into

stochastic gradient descent, which is the

369

method that Posterioris is using, and how

that fits into these different methods

370

that you just talked about?

371

Actually, stochastic gradient descent is

not a method that Posterioris is using per

372

se.

373

descent is stochastic gradient descent is

the workhorse of machine most machine

374

learning algorithms, but posteriors would

kind of be this kind of same like it kind

375

of saying it shouldn't be perhaps or like,

not in all cases.

376

stochastic gradient descent is what you

use.

377

If you have extremely large data, and you

just want to find the MLE or so the

378

maximum likelihood or the minimum of a

loss, which you might say.

379

So

380

that is just as an optimization routine.

381

So you just want to find the parameters

that minimize something.

382

If you're doing variational inference,

what you can do is you can trackively get

383

the KL divergence between your specified

variational distribution and the log

384

posterior.

385

And then you have parameters.

386

So they're like parameters of the

variational distribution over your model

387

parameters.

388

And then you use stochastic gradient

system like that.

389

So this is nice because it just means that

you can throw the workhorse from machine

390

learning at a

391

Bayesian problem and get the Bayesian

approximation out.

392

Again, as we mentioned, it doesn't have

this asymptotic unbiased feature, which is

393

maybe less of a concern in machine

learning models where you have less of

394

ability to trade compute because you've

kind of filled your compute budget with

395

your gigantic model.

396

Although we may see this, we think that I

think this might change over the coming

397

years.

398

But yeah, maybe not.

399

Maybe we'll just go even bigger and bigger

and bigger.

400

You...

401

Okay, sorry.

402

I got lost.

403

You said you're asking about stochastic

gradient descent.

404

So actually, there's something interesting

to say here.

405

And then that means also what the main

difference characteristics of posterior

406

is, like these, so that really people

understand the use case of posterior here.

407

Yeah.

408

So we didn't want to...

409

Okay.

410

So yeah, there's a key thing about the way

we've written posterior is that we like...

411

where possible to have stochastic gradient

descent, so optimization, as sort of

412

limits under some hyperparameter

specifications of the algorithms.

413

And it turns out that in a lot of cases,

so we talked about MCMC, and then we

414

talked about stochastic gradient MCMC,

which are MCMC methods that strictly

415

handle mini -batch methods.

416

And a lot of the time, you can write down

the temperature, you have the temperature

417

parameter of your posterior distribution.

418

And then as you take that to zero,

419

So the temperature is like, if the

temperature is very high, your posterior

420

distribution is very heated up.

421

So you've increased the tails and there's

a lot like a much closer to sort of a

422

uniform distribution.

423

You take it very cold, it comes very

pointed and focused around optima.

424

So we write the algorithms so that there's

this convenient transition through the

425

temperature.

426

So you set the temperature to zero, you

just get optimization.

427

And this is a key thing about posteriors.

428

So we have the, so the posteriors

stochastic grain MCMC methods.

429

this temperature parameter which if you

set to zero will become a variant of

430

stochastic gradient descent.

431

So you can just sort of unify gradient

descent and stochastic gradient MCMC and

432

it's nice so you have your yeah you have

your Langevin dynamics which tempered down

433

to zero just becomes vanilla gradient

descent you have underdamped Langevin

434

dynamics or stochastic gradient HMC,

stochastic gradient Hamiltonian Monte

435

Carlo, you set the temperature to zero and

then you've just got stochastic gradient

436

descent with momentum.

437

So yeah, this is a nice thing about

Posterius to sort of unify these

438

approaches and it hopefully will make it

less scary to use Bayesian approaches

439

because you know you always have gradient

descent and you can sanity check by just

440

setting the temp, just filling with a

temperature parameter.

441

Okay, that's really cool.

442

Okay.

443

So it's like, it's a bit like the

temperature parameter in the, in the

444

transformers that, that like make sure, I

mean, in the LLMs that

445

It's like adding a bit of variation on top

of the prediction stat that the LL could

446

make.

447

Yeah, so it's exactly the same as that.

448

So when you use this in language models or

natural language generation, you

449

temperature the generative distribution so

that the logits get tempered.

450

So if you set the temperature there to

zero, you get greedy sampling.

451

But we're doing this in parameter space.

452

So it's, yeah.

453

It has this, yeah, exactly.

454

Distribution tempering is a broad thing,

particularly in, I'm not going to go too

455

philosophical, but I mean, I've first met

with like tempering, then we thought about

456

it in the settings of sequential Monte

Carlo, and it's like, is it the natural

457

way?

458

Is it something that's natural to do?

459

But in the context of Bayes, because

Bayes' theorem is multiplicative, right,

460

you have your P of theta, P of y given

theta, it kind of makes sense to temper

461

because it means like, okay, I'll just

introduce the likelihood a little bit.

462

and sort of tempering as a natural way to

do it because there's multiplicative

463

feature of Bayes' theorem.

464

So, I kind of settled with me after

thinking about it like that.

465

Yeah, no, I mean, that makes perfect

sense.

466

And I was really surprised to see that was

used in LLMs when I first read about the

467

algorithms.

468

And I was pleasantly surprised because

I've worked a lot on electoral forecasting

469

models.

470

That's how I were introduced to Bayesian

stats.

471

Actually, I've done that without knowing

it.

472

So first I'm using the softmax all the

time because they're called forecasting.

473

Unless you're doing that in the U S you

need a multinomial likelihood.

474

The multinomial needs a probability

distribution.

475

And how do you get that from the softmax

function, which is actually a very

476

important one in the LLM framework.

477

And, and, and also the thing is your

probability is not, it's like the latent.

478

observation of popularity of each party,

but you never observe it, right?

479

And so the polls, you could, you could

like conceptualize them as a tempered

480

version of the true latent popularity.

481

And so that was really interesting.

482

I was like, damn, this like, this, this

stuff is much more powerful than what I

483

thought, because I was like applying only

on electoral forecasting models, which is

484

like a very niche application, you could

say of these models in

485

actually there are so many applications of

that in the wild.

486

No, it's so yeah, tempering in general is

very widespread and also I would say not

487

particularly understood that well.

488

Like yeah, we have this thing, there's

been research in this cold posterior

489

effect which is quite a, I don't know,

it's perhaps a...

490

annoying things for Bayesian modeling on

neural networks where you get, as I said,

491

you have this temperature parameter that

transitions between optimization and the

492

Bayesian posterior.

493

So zero is optimization, one is the

Bayesian posterior.

494

And empirically, we see better predictive

performance, which is a lot of time we

495

care about in machine learning, with

temperatures less than one.

496

So like, yeah, which is annoying because

we're Bayesians and we think that the

497

Bayesian posterior is the optimal decision

-making under uncertainty.

498

So this is annoying, but at least in our

experiments, we found this to be this so

499

-called cold posterior effect much more

prominent under Gaussian approximations,

500

which we only believe to be very crude

approximations to the posterior anyway.

501

And if we do more MCMC or deep ensemble

stuff, where deep ensemble is, we've got a

502

paper we'll be able to archive shortly,

which describes deep ensembles.

503

In deep ensembles, you just run gradient

descent in parallel.

504

with different initializations and batch

shuffling.

505

And then you just have like, I know you

run 10 ensembles, 10 optimizations in

506

parallel, then you've got 10 parameters at

the things at the end.

507

So Monte Carlo approximation posterior

size 10.

508

And then we describe in the paper that how

to get this asymptotic and biased property

509

by using that temperature.

510

Because as we said earlier, you have SG

MCMC becomes SGD with temperature zero.

511

So you can reverse this.

512

for deep ensembles, so you add the noise

and then you'll get an asymptotic and

513

biased deep ensembles become

asymptotically unbiased MCMC between SGMC

514

and PSE.

515

But in those cases when you have the non

-Gaussian approximation we found much less

516

of the cold posterior effect.

517

So yeah, it's, but it's still not, maybe

the cold posterior effect is a natural

518

thing because it's not really like Bayes'

theorem.

519

Yeah, we still need to be better

understood.

520

I don't, at least in my head I'm not.

521

fully clear on whether the cold posterior

effect is something we should be surprised

522

about.

523

Okay, yeah.

524

Yeah, me neither.

525

That makes you feel any better because I

just learned about that.

526

So yeah, I don't have any strong opinion.

527

Okay, I think we're getting clearer now on

the like the what posterior ears is for

528

listeners.

529

So then I think one of

530

the last question about the algorithms

that that's underlying all of that.

531

So, stochastic gradient MCMC.

532

That's, that's where I got confused.

533

Like I hear stochastic gradient and like

stochastic gradient isn't, but no, it's

534

like SG MCMC not SGG.

535

So, Posteriority is like really to use SG

MCMC.

536

Why, like, why would you do that and not

use MCMC?

537

like the classic HMC from Stan or PyMC?

538

Yeah, so I mean, it's not just for SGMCMC.

539

There's also variational inference,

Laplace approximation, extended count

540

filter, and we're really excited to have

more methods as well as we look to

541

maintain and expand the library.

542

Why would you use SGMCMC?

543

So yeah, I think we've already touched

upon this.

544

The thing is, if you've got loads of data,

it's just going to be inefficient to...

545

sum over all of that data at every

iteration of your MCMC algorithm as Stan

546

would do.

547

But there's mathematical reasons why you

can't just do that in Stan.

548

It's because the Metropolis -Hastings

ratio has this exponential of the log

549

posterior.

550

But it's in log space is the only place

you can get the unbiased approximation,

551

which is what you need if you did want to

naively subsample.

552

So you need to, you can't do the

Machrofist Hastings except reject.

553

So you have to use different toolage.

554

And in its simplest terms, SGMCMC just

omits it and just runs a Langevin.

555

So it just runs your Hamiltonian Monte

Carlo without the extract project.

556

But there's more theory on top of this and

you need to control the disqualification

557

error and stuff like that.

558

And I won't go into the weeds of that.

559

Okay.

560

Yeah.

561

Okay.

562

And that's

563

And that's tied to mini -batching

basically.

564

Like the power that SGMCMC allows you when

you're in a high data regime is tied to

565

the mini -batching, if I understand

correctly.

566

It's the difference between MCMC and

SGMCMC.

567

Okay, so that's like the main difference.

568

Okay.

569

Yeah, stochastic gradient.

570

So you can't actually get the exact

gradient like you need in Amazigh in Monte

571

Carlo and for Metropolis Hastings step,

you only get an unbiased approximation.

572

And then there's theory about this is like

sometimes you can deploy the central limit

573

theorem and then you've got a you can go

covariance attached to your gradients and

574

you could do nice theory and improve the

equivalence like that, which, yeah.

575

Okay.

576

All clear now.

577

All clear.

578

Awesome.

579

Yeah.

580

And I think that's the first time we talk

about that on the show.

581

So I think it was it's definitely useful

to be extra clear about that.

582

And so that listeners understand and me,

like myself, so that I understand.

583

Thanks a lot.

584

It's in some setting actually much simpler

because you kind of like remove the tools

585

that you have available to you by removing

that much of the step.

586

So it makes the implementation a bit

simpler.

587

But you kind of lose the theory in that.

588

And then a lot of the argument is like if

you use a decreasing step size, then your

589

noise from the mini match, your noise from

the stochastic gradient decreases Epsilon

590

squared, which is faster.

591

So you

592

If you decrease your step size and run it

for infinite time, then you'll just be

593

running, eventually just be running the

continuous time dynamics, which are exact

594

and do have the right stationary

distribution.

595

So if you run it with decreasing step

size, then you are asymptotically

596

unbiased.

597

But running with decreasing step size is

really annoying because you then don't

598

move as far.

599

As we know from normal MCMC, we want to

increase our step size and move and

600

explore the posterior more so.

601

There's lots of research to be done here.

602

I hope and I feel that it's not the last

time you'll talk about stochastic gradient

603

MCMC on this podcast.

604

Yeah, no.

605

I mean, that sounds super interesting.

606

I'm really interested also to really

understand the difference between these

607

algorithms.

608

Right now, that's really at the frontier

of research.

609

You not only have a lot of research done

on how do you make HMC more efficient, but

610

you have all these new algorithms.

611

approximate algorithms as we said before.

612

So, VLM plus approximation, stuff like

that.

613

But also now you have normalizing flows.

614

We talked about that in episode 98 with

Marilou Gabrié.

615

Marilou Gabrié, actually, I don't know why

I said the second part with the Spanish.

616

Because my Spanish is really available in

my brain right now.

617

So, she's French.

618

So, that's Marilou Gabrié.

619

Episode 98, it's in the show notes.

620

Episode 107, I already mentioned it with

Marvin Schmidt about amortized patient

621

inference.

622

Actually, do you know about amortized

patient inference and normalizing flows?

623

I know a bit about normalizing flows.

624

Amortized patient inference I would be

less comfortable with.

625

Okay.

626

But I mean, if you could explain it.

627

Yeah, I haven't watched that episode and

listened to that episode.

628

Yeah, I mean, we released it yesterday.

629

Yeah, I don't...

630

I'm a bit disappointed, Sam, but that's

fine.

631

Like, it's just one day, you know.

632

If you listen to it just after the

recording, I'll forgive you.

633

That's okay.

634

No, so, kidding aside, I'm actually

curious to hear you speak about the

635

difference between normalizing flows

636

and SGMCMC.

637

Can you talk a bit about that if you're

comfortable with that?

638

I mean, I can't.

639

It's been a while since I've read about

normalizing flows.

640

When I did read about them, I understood

it to be essentially a form of variational

641

inference where you have more elaborate,

you define a more elaborate variational

642

family through like, essentially through

like a triangular mapping.

643

Like, the thing why you can't just use

someone might say,

644

Why can't you use it just a neural network

as your variational distribution?

645

And it's not so easy because you need to

have this tractable form.

646

Hang on a second.

647

Let me remember.

648

But the thing is with normalizing flows,

you can get this because you can invert.

649

That's it.

650

They're invertible, right?

651

Normalizing flows are invertible.

652

So you can get this.

653

You can write the change of distribution

formula and then you can calculate

654

essentially just y -maxum likelihood.

655

the using these normalizing flows to fit

to a distribution.

656

Whereas SGMCMC doesn't.

657

So you have to, in normalizing flows, you

kind of have to define your ansatz that

658

will fit to your distribution.

659

I think normalizing flows are really

exciting and really interesting, but yeah,

660

you have to specify your ansatz.

661

So it's another, so there's another tool

on top, another specification on top of

662

how you.

663

rather than just writing the log

posterior, you then need to find an

664

approximate ansatz which you think will

fit the posterior or the distribution

665

you're targeting.

666

Whereas SGMCMC is just log posterior, go.

667

Which is sort of what we're trying to do

with posterior, is we're trying to

668

automate, well not automate, we're trying

to research, of course, so much for that.

669

But normalizing flows might be, yeah, as I

said, I think it's really interesting that

670

you can get these more expressive

variational families through like

671

triangular mappings, yeah.

672

Yeah, super interesting.

673

And yeah, I'm also like spatial inference

is related in the sense that you first

674

feed a deep neural network on your model.

675

And then once it's feed, you get posterior

inference for free, basically.

676

So that's quite different from what I

understand as GMC to be.

677

But that's also extremely interesting.

678

That's also why I'm

679

hammering you down on the different use

cases of SGMCMC so that myself and

680

listeners have a kind of a tree in their

head of like, okay, my use case then is

681

more appropriate for SGMCMC or, no, here

I'd like to try multi -spacian inference

682

or, I know here I can just stick to plain

vanilla HMC.

683

I think that's very interesting.

684

But thanks for that question that was

completely improvised.

685

I definitely appreciate you taking the

time to rack your brain about the

686

difference with normalizing flows.

687

No, I'd love to talk more on that.

688

I'd need to refresh myself.

689

I've written down some notes on

normalizing flows, and I was quite

690

comfortable with them, but it's just been

a while since I refreshed.

691

So I would love to refresh, and then we

can chat about them.

692

Because I'd love to do a project on them,

or I'd love to work on them, because I

693

think that's it.

694

way to fit distribution to data, which is,

after all, a lot of what we do.

695

Yeah.

696

Yeah.

697

So that makes me think we should probably

do another episode about normalizing

698

flows.

699

So listeners, if there is a researcher you

like who does a lot of normalizing flows

700

and you think would be a good guest on the

show, please reach out to me and I'll make

701

that happen.

702

Now let's let's get you closer to home

salmon and talk about posteriors again

703

Because so basically if understood

correctly posteriors aims to address

704

uncertainty quantification in deep

learning Why it's is that my right here

705

and also if that's the case why is this

particularly important for neural networks

706

and How does the package help in?

707

managing especially overconfident in model

predict, overconfidence in model

708

predictions.

709

Yeah, so it's that's our primary use case.

710

And normal is to use posterity as a

proximate base, we're getting as close to

711

base as we can, which is probably not that

close, but still getting somewhere on the

712

way to base base, base and posterior in

big deep learning models.

713

But we feel posterior is to be as modular

and general as possible.

714

So as I said, if you have a

715

classical Bayesian model, you can write it

down in Pyro, but you've got loads of

716

data, then okay, go ahead.

717

And it posterior should be well suited to

that.

718

In terms of what advantages we want to see

from uncertainty communication or this

719

approximate Bayesian inference in deep

learning models, there are three sorts of

720

key things that we distilled it down to.

721

So yeah, you mentioned

722

confidence in outer distribution

predictions.

723

So yeah, we should be able to improve our

performance in predicting on inputs that

724

we haven't seen in the training set.

725

So I'll talk about that after this.

726

The second one is continual learning,

where we think that if you can do Bayes

727

theorem exactly, you have your prior, you

get some likelihood and you have the

728

likelihood, you get some data, you have a

posterior, then you get some more data.

729

and then your posterior becomes your prior

and do the update.

730

And you can just write like that if you

can do Bayes' theorem exactly.

731

And then, yeah, this is, you can extend it

even further and then you have, with some

732

sort of evolution along your parameters,

then you have a state space model, and

733

then the exact setting linear Gaussian,

you've got a count filter.

734

So continual learning is, in this case,

Bayes' theorem does that exactly.

735

And in continual learning research in

machine learning settings, they have this

736

term of avoiding catastrophic forgetting.

737

So,

738

If you just continue to do gradient

descent, there was no memory there, so you

739

would just, apart from the initialization,

you would just forget what you've done

740

previously and there's lots of evidence

for this, whereas Bayes' theorem is

741

completely exchangeable between of the

order of the data that you see.

742

So you're doing Bayes' theorem exactly,

there's no forgetting, you just have the

743

capacity of the model.

744

So that's where we see Bayes solving

continual learning, but as I said, you

745

can't

746

can't do Bayes' theorem exactly in a

billion -dimensional model.

747

And then the last one is, we'll call it

like decomposition of uncertainty in your

748

predictions.

749

So if you just have gradient descent model

and you're predicting reviews, someone's

750

reviews and you have to predict the stars,

it will just give you, as you said, it

751

gives you your softmax, it'll just give

you this distribution over the reviews and

752

it'll be like that.

753

But what you really want is you want to

have some indication of

754

like also distribution detection, right,

you want to know, okay, yeah, I'm

755

confident in my, my prediction.

756

And you might get to review that is like,

the food was terrible, but the service was

757

amazing, or something like that, like a

user amazing food was terrible.

758

And then, like, let's say we're perfect

models, say of this, we know how people

759

review things, but we can we can give, we

have quite a lot of uncertainty under

760

review, because we don't know how the

reviewer values those different things.

761

So we might have just a completely

uniform.

762

distribution over the stars for that

review.

763

But we'd be confident in that

distribution.

764

But what Bayes gives you is it gives you

the ability to the sort of the second

765

order uncertainty quantification, is if

you have this distribution over parameters

766

and you have a distribution over logits at

the end, the predictions, you can

767

identify, you can split between it from

information theories called aleatoric and

768

epistemic uncertainty.

769

Aleatoric uncertainty or data uncertainty

is what I just described there, which is

770

natural uncertainty in the model and the

data generating process.

771

Epistemic uncertainty is uncertainty that

was removed in the infinite data limit.

772

So that would be where the model doesn't

know.

773

So this is really important for us to

quantify that.

774

Okay.

775

I, yeah, around to the bit there.

776

I can in like 30 seconds elaborate on the

point you specifically mentioned on alpha

777

distribution performance and improving

performance and alpha distribution.

778

And I think that's quite compelling from a

Bayesian point of view, because what Bayes

779

says on like a supervised learning sector

setting is said, gradient descent just

780

fits one parameter, finds one parameter

configuration that's plausible given the

781

training data.

782

Bayes' theorem says, I find the whole

distribution of parameter configurations

783

that's plausible given the data.

784

And then when we make predictions, we

average over those.

785

So it's perfectly natural to think that a

single configuration might overfit.

786

and might just give, it might just be very

confident in its prediction when it sees

787

out the distribution data.

788

But it doesn't necessarily solve a bad

model, but it should be more honest to the

789

model and the data generating process

you've specified is if you average over

790

plausible model configurations under the

training data when you have your testing.

791

So that's sort of quite a compelling, to

me, argument for improving

792

performance on after distribution

predictions, like the accuracy of them.

793

And there's a fair bit of empirical

evidence for this, with the caveat again,

794

being that the Bayesian posterior in high

dimensional models, machine learning

795

models is pretty hard to approximate, cold

posterior effect, caveats, things like

796

these things.

797

Okay, yeah, I see.

798

Yeah, super interesting in that.

799

So now I understand better.

800

what you have on the posteriors website,

but the different kind of uncertainties.

801

So definitely that's something I recommend

listeners to give a read to.

802

I put that in the show notes.

803

So both your blog post introducing

posteriors and the docs for posteriors,

804

because I think it makes that clear

combined to your explanation right now.

805

Yeah.

806

And...

807

Something I was also wondering is that if

I understood correctly, the package is

808

built on top of PyTorch, right?

809

Yeah, that's correct.

810

Yeah.

811

Okay.

812

So, and also, did I understand correctly

that you can integrate posteriors with pre

813

-trained LLMs like Lama2 and Mistral, and

you do that with a...

814

Hacking's Feast Transformers package?

815

So, yeah, so, I mean, yeah, Posterior is

open source.

816

We're fully supported the open source

community for machine learning, for

817

statistics, which is, and in terms of,

yeah, I mean, we're sort of in the fine

818

tuning era or like we have like, there's

so much, there are these open source

819

models and you can't get away from them.

820

We have that Lama 2, Lama 3, Mistral,

like, yeah.

821

And basically we want to harness this

power, right?

822

But as I mentioned previously, there are

some issues that we like to remedy with

823

Bayesian techniques.

824

So the majority of these open source

models are built in PyTorch.

825

I'm also a big Jax fan.

826

I also use Jax a lot.

827

So I was very happy to see and work with

the torch .funk like sub library, which

828

basically makes it

829

you can write your PyTorch code and you

can use Llama 3 or Mistral with PyTorch

830

but writing functional code.

831

So that's what we've done with Posterior.

832

So, yeah, Hugging Face Transformers, you

can download the models, that's where all

833

they're hosted, and how you access them.

834

But then what you get is just a PyTorch

model.

835

It's just a PyTorch model.

836

And then you throw that in Composers and

all nicely with the Posterior updates.

837

Or you write your own new updates in the

Posterior framework and you can use that

838

as well.

839

still with Lama 3.

840

Mr.

841

Robin.

842

Yeah.

843

Okay.

844

Nice.

845

And so what does it mean concretely for

users?

846

That means you can use these pre -trained

LLMs with posteriors and that means adding

847

a layer of uncertainty quantification on

top of those models?

848

Yeah.

849

So you need, I mean, Bayes theorem is a

training theorem.

850

So you need data as well.

851

So you take

852

You take your pre -trained model, which

is, yeah, transformer, or it could be

853

another type of model, it could be an

image model or something like that, and

854

then you give it some new data, which we

would say was fine -tuning, and then you

855

combine, use posterior to combine the two,

and then you have your new model out at

856

the end of the day, which has uncertainty

quantification.

857

It's difficult, as I said, we're sort of

in this fine -tuning era as open -source

858

large language models.

859

It's still to be, this is different.

860

There's still lots of research to do here

and it's different to our classical

861

Bayesian regime where we just have our,

there's only one source of data and it's

862

what we give it.

863

In this case, there's two sources of data

because you have your data, whatever,

864

whatever Lama3 saw in its original

training data and then it has your own

865

data.

866

It's, yeah, can we hope to get uncertainty

chronification and the data that they used

867

in the original training?

868

Probably not, but we might be able to get

uncertainty chronification and improved

869

predictions.

870

based on the data that we've committed.

871

So there's lots of lots for us to try out

here and learn because we are still

872

learning on this in terms of the fine

tuning.

873

But yeah, this is what Polastir is there

to make these sort of questions as easy as

874

possible to ask and answer.

875

Okay, fantastic.

876

Yeah, that's, that's so exciting.

877

It's just like, it's a bit frustrating to

me because I'm like, I'd love to try that

878

and learn on that and like, contribute to

that kind of packages.

879

At the same time, I have to work, I have

to do the podcast, and I have all the

880

packages I'm already contributing to.

881

So I'm like, my god, too much choices too

much, too many choices.

882

No, come on Alex, I'm gonna see you.

883

We're gonna see you again, Alex pull

request.

884

It's soon enough.

885

Actually, how does like, do the like this

ability to have the transformers in, you

886

know, use these pre trained models, does

that help facilitate the adoption of new

887

algorithms in in posteriors?

888

Because if I understand correctly, you can

support

889

new algorithms pretty easily and you can

support arbitrary likelihoods.

890

How do you do that?

891

I wouldn't say that the existence of the

pre -trained models necessarily allows us

892

to support new algorithms.

893

I feel like we've built the posterior to

be suitably general and suitably modular,

894

that it's kind of agnostic to your model

choice and your log posterior choice.

895

terms of arbitrary likelihoods.

896

But yeah, that's like a benefit.

897

That's like, yeah, as an hour, yeah, the

arbitrary like is is relevant, because a

898

lot of machine learning packages on.

899

I mean, a lot of machine learning is

essentially just boils down to

900

classification or regression.

901

And that is true.

902

And because of that, a lot of a lot of

machine learning algorithms will a lot of

903

machine learning packages will essentially

constrain it to classification or

904

regression.

905

At the end, you either have your softmax

or you have your mean squared error.

906

Yeah, softmax cross entry means greater.

907

In posterior, we haven't done that.

908

We're more faithful to the sort of the

Bayesian setting where you just write down

909

your log posterior and you can write down

whatever you want.

910

And this allows you greater flexibility in

the case you did want to try out a

911

different likelihood or like even in like

simple cases, like it's just more

912

sophisticated than just classification or

regression a lot of the time.

913

Like in sequence generation where you have

the sequence and then you have the cross

914

entropy over all of that.

915

It just allows you to be more flexible and

write the code how you want.

916

And there's additional things to be taken

into account.

917

Like sometimes if you were doing a

regression, you might have knowledge of

918

the noise variance.

919

And that's just the observation noise

variance.

920

And that's just much easier to, yeah, if

we don't constrain like this, it's just

921

much easier to write your code much

cleaner code than if you were.

922

And it's also future -proofing.

923

We don't know what's going to be.

924

happening in going forward.

925

We may see like, yeah, in multimodal

models, we may see like, text and images

926

together, in which case, yeah, we will

support that.

927

You have to supply the compute and the

data, which might be the harder thing, but

928

we'll support those likelihoods.

929

Okay, I see.

930

I see.

931

Yeah, that's very, very interesting.

932

Any stats related to the fact that I think

I've read in your blog post or on the

933

website that

934

You say that Posterior is swappable.

935

What does that mean?

936

And how does that flexibility benefit

users?

937

Yeah.

938

So, I mean, this is the point of swappable

is that when I say that is that you can

939

change between if you want to, if you

think, as I said, Posterior is a research

940

like toolbox and it's to us to investigate

which inference method is appropriate in

941

the different settings, which might be

different if you care about decomposing.

942

predictive uncertainty, it might be

different if you care about boarding cast

943

-off, you're forgetting it's in your

continued learning.

944

So the thing is that you can just, the way

it's written is you can just swap, you can

945

go from sthmc and you can go to the class

approximation or you can go to vi just by

946

changing one line of code.

947

And the way it works is like you have your

builds, you have your transform equals

948

posterior .infant method .build and then

any configuration argument, step size.

949

things like this, which are algorithm

specific.

950

And then after that is all unified.

951

So you just have your init around the

parameters that you want to do based on.

952

And then you iterate through your data

loader, you iterate through your data.

953

And then it just updates based on the

batch.

954

And batch can be very general.

955

So that's what it means.

956

So you can just change one line of code to

swap between Variational Imprints and

957

STHMC or Extended Calama Filter or any and

all the new methods that the listeners are

958

going to add in the future.

959

Heh.

960

Okay.

961

Okay.

962

I see.

963

And so I have so many more questions for

you and posterior's but let's start and

964

run, wrap that up because also when I ask

you about another project you're working

965

on so maybe to close that up on

posterior's.

966

What are the future plans for posterior's

and are there any upcoming features or

967

integration integrations that you can

share with us?

968

So we're quite happy with the framework at

the moment.

969

There's lots of little tweaks that we have

a list of GitHub issues that we want to go

970

through, which are mostly and excitingly

about adding new methods and new

971

applications.

972

So that's really what we're excited about

now is actually use it in the wild and

973

hopefully experiment all these questions

that we've discussed.

974

Yeah, like, like how we how does it make

sense and how we get the benefits of

975

Bayesian, true Bayesian inference on fine

tuning or on large models or large data.

976

And so yeah, we are really excited and to

add more methods.

977

So if listeners have mini batch, big data

Bayesian methods that we want to want to

978

try out with a large data model, then

we're hopefully accepting that we will.

979

I do.

980

I do like, I do promote like generality

and doing it like in a way that is sort of

981

flexible and stuff.

982

So we may have, we may think a lot.

983

It's not, it's not, we want to add methods

that somehow feel natural and, and one way

984

is to extend and compose with other

methods.

985

So it might be that if we've got some very

complicated last layer,

986

requires classes just for classification

method, we're probably not going to add

987

it.

988

So it has to be methods that stick within

the posterior framework, which is this

989

arbitrary likelihood Bayesian swappable

computation.

990

Okay.

991

Okay.

992

Yeah.

993

Yeah, that makes sense.

994

Yeah, because you have like, yeah, you

have that kind of vision of wanting to do

995

that and having that as a as a research

tool, basically.

996

So

997

Yeah, that makes sense to keep that under

control, let's say.

998

Something I want to ask you in the last

few minutes of the show is about

999

thermodynamic compute.

I've seen you, you are working on that.

And you've told me you're working on that.

So yeah, I don't know anything about that.

So can you like, what's that about?

Yeah, so I mean, this is yeah, this is

something that's very normal, normal

computing.

And it's like,

It's something that we have.

Yeah, we have this hardware team.

It's like a full stack AI company.

And we, yeah, on the posterior side, on

the client side, we look at how we can

bring in principle Bayesian uncertainty

quantification and help us solve the

issues with machine learning pipelines

like we've already discussed.

And on the other side, there's lots of

parts to this.

More just like traditional MCMC is

difficult sometimes because

Or just it's just like about simulating

SDEs essentially as what the thermodynamic

hardware is simulating SDEs Normally, you

have this real pain with the step size and

as the mention grows steps, let's get

really small and so SDEs, where do we see

SDEs?

You see SDEs in physics all the time and

physics is real we can use physics so it's

doing so it's building physical hardware

analog hardware that We can hopefully that

evolves as SDEs

then we can harness that SDEs by encoding,

you know, like currents and voltages and

things like that.

So I'm not a physicist, so I don't know

exactly how it is.

But I'm always reassured at how the when I

speak to the hardware team, how simple the

they talk about these things, it's like,

yeah, we can just stick some resistors and

capacitors on a chip, and then it'll then

it'll do this SDE.

So this is the and then we want to use

those SDEs for scientific computation.

And with a real focus on statistics and

machine learning.

So yeah, we want to be able to do an HMC

on device, on an analog device.

The first step is to do like with a

linear, so we'll have a Gaussian posterior

or with a linear drift in terms of this.

This is an Ornstein -Ollenbeck process and

we've developed hardware to do this and

turns out that an Ornstein -Ollenbeck

process, because it has a Gaussian

stationary distribution and you have this,

you can input like you can input the

precision matrix and output the covariance

matrix, that's matrix inversion.

So, and you just, your physical device

just does this.

And it's because it's an SDE, it has noise

and is kind of noise aware, which is

different to classical analog computation,

which has historically been plagued, which

is really old, really old, but

historically been plagued by noise.

And it's like, yeah, there's all this

noise in physics.

And because we're doing SDEs, we want the

noise.

So yeah, that's the whole idea.

It's obviously very young, but it's fun.

It's fun stuff.

Yeah.

So that's basically to...

accelerate computing?

That's hardware first, so that computing

is accelerated?

We want to, I mean, it's a baby field.

So we're trying to accelerate different

components.

What we worked out is with the simplest

thermodynamic chip we can build is this

linear chip with the Ornstein -Ullenberg

process.

And that can speed up with some error.

some error, but it has asymptotic speed

ups for linear algebra routines, so

inverting a matrix or solving a linear

system.

That's awesome.

In this case, it would speed up a certain

component, but that could be useful in a

Laplace approximation or these sort of

things also in machine learning.

Okay, that must be very fun to work on.

Do you have any writing about that that we

can put in the show notes?

Because

I think it'd be super interesting for

listeners.

Yeah, yeah.

We've got the normal computing scholar

page has a list of papers, but we also

have more accessible blogs, which I'll

make sure to put in the shop.

Yeah, yeah, please do because, yeah, I

think it's super interesting.

And yeah, and when you have something to

present on that, feel free to reach out.

And I think that'd be fun to do an episode

about that, honestly.

That'd be great.

Yeah.

Yes, so maybe one last question before

asking you the last two questions.

Like very, like, let's do Zoom be way less

technical.

We've been very technical through the

whole episode, which I love.

But maybe I'm thinking if you have any

advice to give to aspiring developers

interested in contributing to open source

projects like Posterior's, what would it

be?

Okay, yeah, I don't know, I don't feel

like I'm necessarily the best place to say

all this, but yeah, I mean, I would just,

the most important thing is just to go for

it, just get stuck in, get in the weeds of

these libraries and see what's there.

And there's loads of people building such

cool stuff in the open source ecosystem

and it's really fun to, honestly, it's

really fun and rewarding to get involved

for it.

So just go for it, you'll learn so much

along the way.

something more tangible.

I find that when I'm stuck on, starting

on, it's not like I don't understand

something in code or mathematics, then I

often struggle to find it in papers per

se.

And I find that textbooks, I love

textbooks, textbooks I find as a real

source of gold for these because they

actually go to the depths of explaining

things, without having this sort of horse

in the race style writing that you often

find in papers.

So yeah, get stuck in check text textbooks

if you, if you get lost.

Or I don't understand.

Or just ask as well.

Open source is all about asking and

communicating and bouncing ideas.

Yeah, yeah, yeah, for sure.

Yeah, that's usually what I do.

I ask a lot and I usually end up

surrounding myself with people way smarter

than me.

And that's exactly what you want.

That's exactly how I learned.

Yeah, textbook DICI, I would say I kind of

find the writing boring most of the time,

depends on the textbooks.

And also, it's expensive.

Yeah.

So that's kind of the problem of

textbooks, I would say.

I mean, you often can have them in PDFs,

but I just hate reading the PDF on my

computer.

So, you know, I wonder on the book object

or having it on Kindle or something like

that.

But that doesn't really that doesn't

really exist yet.

So.

could be something that some editors solve

someday that'd be cool, I'd love that

awesome, Sam, that was great thank you so

much, we've covered so many topics and my

brain is burning so that's a very good

sign I've learned a lot and I'm sure our

listeners did too of course, before

letting you go I'm gonna ask you the last

two questions I ask every guest at the end

of the show so one

If you had unlimited time and resources,

which problem would you try to solve?

want to decouple the model specification,

the data generating process, how you go

from your something you don't know to the

data you do have.

That's your site freedom as a data model.

I have you define that from like the

inference and the mathematical

computation.

So that's whatever, what the way you do

your approximate Bayesian inference.

And you want to decouple those.

You want to make it as easy as possible.

Ideally, we just want to be doing that

one.

We just want to be doing the model

specification.

And this is like Stan and PyMC do this

really well.

It's just like,

you write down your model, we'll handle

the rest.

And that's kind of like the dream we have

as Bayesian or Bayesian software

developers.

And it's so with Posterior, we're trying

to do something like this towards going to

move towards this for bigger, big machine

learning models and so bigger models,

bigger data settings.

So that's kind of the dream there.

But then in machine learning, what does

machine learning have differently to

statistics in that setting?

It's like, well, machine learning models

are less interesting than classical

Bayesian models.

The thing is they're more transferable,

right?

It's just a neural network, which we

believe is machine learning and will solve

a whole suite of tasks.

So perhaps in terms of the machine

learning setting, where we decouple

modeling and inference and data, you kind

of want to remove the model one as well.

You want to have these general purpose

foundational models, you could say.

So really you want to let the user focus.

And so we're handling the inference.

We're also handling the model.

So really let the user just give it the

data and say, okay, let's do this data and

let's use this data to predict other

things and let the user handle that.

So that's potentially like a real

unlimited time and resources.

Plenty of resources need to do that.

Yeah.

Yeah, that sounds...

That sounds amazing.

I agree with that.

That's a fantastic goal.

And yeah, also that reminds me, that's

also why I really love what you guys are

doing with Posteriorus because it's like,

yeah, trying to now that we start being

able to get there, making patient

inference really scalable to really big

data and big models.

I'm super enthusiastic about that.

it would be just fantastic.