Welcome everyone to this seminar. We have Professor Sebastian Reich from the University

of Prostam, Germany, and he's speaking today about statistical inverse problems and gradient

flow structures in the space of probability measures. Please, Professor Reich.

Yeah, thank you. Thank you for the introduction. It's a pleasure to speak. So, yeah, so my

main area of interest is in what's called data assimilation these days. So it's how to merge

models and data. Reportedly speaking, these are statistical inverse problems. Most of the time,

they're time dependent, but today I will actually primarily focus on time-independent problems

within the Bayesian perspective. So to set the stage, and by the way, if you have any

urgent questions in terms of not understanding something, then please ask immediately. Otherwise,

if I don't forget, I will occasionally take a little break and then we will have time for questions

in between as well. And of course, there's the usual question session at the end, I suppose.

Anyway, so Bayesian inverse problems, you have data, which I will denote by y.

And then there's an unknown parameter you like to estimate. And in my case, these are denoted by x.

And then you basically generally assume that the data is linked, that I simplify it here a little

bit, that the data is linked to the parameters of interest, in this case, linearly. Of course,

generally it's nonlinear, but for simplicity, you just keep it linear. So h is a forward map.

And then you also assume that there are some nodes in the data. And also here, again, for simplicity,

I assume that the noise is Gaussian. So having this model for the data and how it connects to the

variable of interest leads to this data likelihood, which is basically the Gaussian function for this.

And then in Bayesian inference, you also assume that you have some sort of prior knowledge

about this variable of interest. And this is a distribution. So the contrary to a frequentist

approach, the Bayesian approach assumes that the parameter is a random variable. And so you have

this prior assumption, which sort of collects some understanding of the variable of interest.

You have your likelihood. And then Bayes' theorem basically computes the condition of the

variable of interest, even the data. So I will denote this by pA. And the conditional I will

suppress. So typically, you really write in Bayes, you write x conditioned on the data.

But I will, for simplicity, just write it as pA. Now,

f, that's the sort of jargon that comes out of the geosciences. f is like forecast.

And a is analysis. But it's sort of the prior and the posterior in the classical Bayesian context.

So this is the problem we like to tackle. And then of course, the general interest,

question of interest is to compute expectation of the data. And so this is the problem that we

compute expectation values, some sort of summary statistics of this posterior distribution.

Sorry, could you, the forecast and the a, the analysis, could you give us some two sentences

on this? What is the image about the forecast and the analysis?

Okay, so think of weather forecasting, right? So the DWD has a fluid dynamics solver,

including all the whistles and bells for the atmosphere. And basically, you start with the

current day, initialize it, and then run it into the future for next day. And that would be a

forecast. And in fact, the DWD runs it in an ensemble way. So they're not just releasing a

single forecast, but they release with perturbed initial conditions, they release an whole ensemble.

So you get actually a sample of forecasts and ensemble forecasts, right. And that's

to represent the forecast for the next day. That's why the forecasting comes in. And then

the next day, you collect data, right? In the meantime, you keep collecting data.

And that's what this is. And then tomorrow, you want to update your forecast,

when tomorrow becomes today, using the data you've collected in between. So you want to adjust your

forecast. And that's the analysis. So you literally, we will actually talk about this later,

how this is done in the weather forecast, what the DWD does, I will tell you actually.

You adjust these forecasts, and then you use them as initial conditions for your next forecast,

which is then the next day after tomorrow, once tomorrow has become today. So this cycle,

that's where this forecasting analysis jargon comes from. Because you really think of this

as a sequential procedure that you do day by day in terms of weather forecasting. Now,