# How does the prior enter the calculation when estimating the evidence using thermodynamic integration?

I am attempting to perform Bayesian model comparisons. To calculate the evidence, or marginal likelihood, for each model I use so-called thermodynamic integration having fitted the model using a parallel tempered MCMC algorithm.

### Background

The theory of thermodynamic integration is explain in 2.2 of this work and the implementation I'm using is documented here.

The fundamental result is that the evidence, defined as

$$Z = \int_{\theta} P(y|\theta) P(\theta) d\theta$$

where $\mathbf{y}$ is the data and $\theta$ are the model parameters, can be approximated by

$$\log(Z) = \int_{0}^{1} \langle P(y | \theta) \rangle_{\beta} d\beta$$

where $\langle P(y | \theta) \rangle_{\beta}$ is the averaged likelihood at inverse-temperature $\beta$.

### The question

The issue I have is that, it seems to me that $P(\theta)$ has dropped out of the calculation in taking the approximation. So, does the evidence, as calculated via thermodynamic integration, include the evidence?