evidence

This page is on the Bayesian evidence, with the notation $\text{Pr}(D \vert M)$ ; the probability of data $D$ under model $M$ .

In an alternative way of writing Bayes' rule:

$p(\Theta) = \frac{\mathcal{L}(\Theta)\pi(\Theta)}{\mathcal{Z}}$

where $\mathcal{Z}=\int_{\Omega_{\Theta}} \mathcal{L}(\Theta)\pi(\Theta)~d\Theta$ ; i.e. the posterior $p(\Theta)$ is given by the normalized product of likelihood and prior, with $\Theta$ in the hypothesis space $\Omega_\Theta$ . In essence, the posterior is just a vector of weights to $\Theta$ .

The motivation behind this is to be able to compare between models $M$ , which involves integration over volumes of parameter space, as opposed to the surface sampled by conventional MCMC.

One way to do this is using (dynamic) [[nested-sampling]].

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nested-sampling

Sampling bounded shells of the hypothesis space, in order to efficiently compute the Bayesian [[evidence]].

bayes-rule

where $p(y)$ is the [[evidence]], $p(x)$ is the [[prior]], $p(x \vert y)$ is the [[posterior distribution]], and $p(y \vert x)$ is the [[conditional-distribution]] known as the likelihood.