Hi, it's time to talk about Monte Carlo integration and we start by recording what we talked about

when we talked about Bayesian inverse problems.

So we assume that we have some unknown quantity x, which is an unknown parameter or something

you'd like to infer, and this parameter is a priori distributed according to this prior

measure mu0, which has density rho x.

So this is our symbol for density with respect to the Lebesgue measure.

We could also say that rho x is the Radon-Nikodym derivative of mu0 with respect to the Lebesgue

measure.

And there's some measurement noise epsilon, which is also distributed according to some

density which we call rho epsilon.

Now y is given by g of x plus epsilon, so this constitutes a measurement process where

this unknown variable is mapped by some maybe complicated mapping g and the result is then

perturbed by some additive measurement noise epsilon.

And we saw that the distribution of the data given a fixed value of the parameter x is

given by this likelihood function here, which is just the density of the measurement noise

and we plug in y minus g of x.

Then we applied Bayes theorem and that said that we can write down the posterior, which

is the distribution of x given data y.

We call this mu y in order to denote the dependency on the specific value of y.

This measure has the back density rho of x given y is equal to y and that is given by

some constant, which is the normalization times the likelihood and times the prior.

And we saw that when we wrote this as d mu0 divided by dx, then we could put this on the

other side and we got that the Radon-Nikodym derivative of the posterior with respect to

the prior is given by this normalization times the likelihood.

And this c, there's a typo here, this should say 1 over c.

1 over c is the evidence and this is the density of y, which is the same as marginalizing out

the joint density of x and y.

So that's what we saw last time.

Now again with those two different ways of writing the posterior, so either we look at

the posterior in terms of its back density, so this is this line.

This line says we can evaluate the measure of sets A with respect to this mu y by integrating

the back measure.

So this is the proper Lebesgue integral, the usual integral that we look at and we integrate

over the density, this rho y density.

Or we can get the same quantity by not integrating over the back measure but over the prior and

then we have to change the integrand to that function, which of course is that function

here.

So those are two options of looking at the same quantity, two different kinds of integrals

and in some context it will be easier to think about this measure as having a Lebesgue density

and sometimes it will be easier to think about this measure having a prior density.

Okay so now we're given a posterior distribution mu y, so that is the correct way of combining

our prior with the data and let's say this gives us some posterior distribution like

that, so that shows the level sets of the density of this posterior.

So it looks quite complicated, it has multiple peaks, it's skewed and warped and there are

difficult shapes in here, there's kind of an isolated peak here, it's a really complicated

distribution.

So here we can still plot it, so in two dimensions we can look at that, we can say oh there seems

to be a really important region here and important region here, there's kind of an x structure

here and blah blah blah.

So we can kind of do an image analysis here, but in higher dimensions that is not possible.