# Markov Chain Monte Carlo

We are interested in new sampling algorithms, and more interestingly at methods to calibrate the samplers.

## The motivation behind MCMC algorithms

Let us briefly explain where it comes from and the motivations behind it. Let us consider a response variable \(\bb{Y}\), feature matrix \(X\) and a model with parameter \(\boldsymbol{\beta}\) so the likelihood of the data given the model and parameter values is given by

\[ P\left(Y | X, \beta\right) \]

Given a test point \(x_*\) we are usually interested in the distribution over predictions \(Y_*\) (a random variable):

\[ P\left(Y_* | Y, X, x_*\right) \]

Let us a assume we have a function (assumed to be deterministic here) that returns a sample \(y_*\) of \(Y_*\)'s distribution given a value of the set of parameters \(\tilde{\beta}\):

\[ y_* = f(\tilde{\beta}, x_*) \]

Then we get this by marginalizing:

\[ P\left(Y_* | Y, X, x_*\right) = \int P(f\left(x_*, \beta\right)| \beta, x_*) P(\beta|Y, X) \mathrm{d} \beta \]

where \(P(\beta|Y, X)\) is the *posterior distribution* of the model's parameters. This is the integral that we would like to evaluate for all practical purposes. Given \(\left\{\tilde{\beta}_1, \dots, \tilde{\beta}_N \right\}\) \(N\) samples from the posterior distribution.

(Adrien) - Should look at Ensemble MCMC.

(Junpeng) - Delayed rejection sampling, Hamiltonian dynamics with non-newtonian momentum for rapid sampling