Markov chain Monte Carlo

Um, and then you observe some data, and then you want to, um, reason about the posterior over a function or the posterior over parameters that describe the function.

And, um, for computing that posterior, you can use Markov chain Monte Carlo, or you can use variational inference, and both of them have their issues.

MCMC is just really slow, and variational inference is, well, there's complex math, and the approximations sometimes, um, don't work out perfectly, and it's also sometimes a bit slow.

And, um, but then once you have this, uh, posterior over, um, functions or parameters, then typically in order to get the posterior predictive distribution, the P of Y given X and the data, that is actually typically easy.

Um, if you have a sample-based approximation of the posterior, um, over functions, then yeah, you-- that, that's basically just a sum over, um, that sample-based approximation.

So then people went and said, well, we can do a little bit better by framing this as basically as the limiting process of a Markov process.

And this gives us something MCMC, where basically the idea is that we may not be able to create the density, but we are able to sample from it.

And that's one of the very interesting things about everything inference or density-based, is that you can actually do a lot of things without actually knowing the density itself, just by sampling and rejecting samples.

The issue with this is that if you've ever run like Markov to Monte Carlo, you will notice this is insanely slow, especially in high dimensions.

So at some point, energy-based models popped up and these normalizing flows.

Explain that to me.

Yeah, so energy-based models actually directly come from MCMC, right? The idea or like the reason it's called energy is because it comes from physics.

Like in the case of physics, they literally had like a function of energy and then you would assume that kind of the particles kind of conform proportional to the energy, right? Let's say you have a specific...

The idea is that none of this methodology should try to enumerate them.

And there are techniques, especially in the context of the Markov chain Monte Carlo simulation, which are designed to simulate complex distributions actually defined on this combinatorial space.

One of them is called Metropolis Hastings, and this is really designed to actually focus on what matters.

So just to give you an example, even if you have billions of alternatives, Most of them are irrelevant because they are completely off.

So we sample alternative where it matters.

And usually we draw from the actual choice model and and then we correct for the bias we create.

And thanks to this machinery of Markov chain Monte Carlo, we can actually do that.

The last one from Adeline is, do you ever do constant validations tests when you calibrate the modeling using historical data and then try to run it forward to the present to see if the outputs are matching current statistics as a back check step?

You didn't necessarily want to retrain your model for every one of those occasions.

So we were willing to pay a large price to train the likelihood to then make it feasible to do MCMC, right? So we are amortizing the likelihood and then inference goes back to all our normal machinery, right?

So

we can do inference.

We are not basic MCMC, right? If you want variation inference downstream and you're able to just test a lot of different things while having amortized like one thing, right? But we're willing to pay a pretty hefty price to amortize that one thing, right? And the posterior route is quite motivated by having nearly instant inference by the time you have it amortized.

But then especially initially, it means that you're amortizing a very specific scenario, right? So the likelihood route gives you basically all the flexibility downstream for free, but you pay a larger price per inference.

The posterior amortization route makes inference instant, but locked you into particular scenarios.

So you're trying to amortize the inference step, and you can even do that for models for which you a priori have likelihoods, right? You can do that.

that? That's an interesting process for me as well.

So, In insurance, also in financial services, there's this concept called the Markov chain, or we usually say like MCMC models.

So the core principle of these Markov chains is that it can create multiple simulations of the same fact.

So mostly this is used for like, if you are doing like a serious trading or managing like a financial portfolio, including an insurance capital management site, use this specific statistical model to build variations of the underlying model and then test your assumptions on that.

And again, I've not been an academic for a long time, but I remember, and this is super gross, like, you know, overgeneralization, but sort of two types of successful academics.

They're arbitrageurs, right, who go and find a method or something being in a different field and ported in and become pioneers because they were the first one not to do it, but like the first person to use IRT on legislatures made their name on that, right? The first person to apply, you know, some kind of Bayesian MCMC estimation to a different problem.

And IO made their

name on