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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?

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