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 $\text{bb}Y$, feature matrix $X$ and a model with parameter $\mathbf{\beta }$ 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 $x_{\ast }$ we are usually interested in the distribution over predictions $Y_{\ast }$ (a random variable):

$$P\left(Y_{\ast }|Y,X,x_{\ast }\right)$$

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

$$y_{\ast }=f(\tilde{\beta },x_{\ast })$$

Then we get this by marginalizing:

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

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

References

• Monte Carlo integration (Wikipedia)
• Andrieu et al A general perspective on the Metropolis-Hastings kernel (2020)