Bayes Rule with Model Comparison In Doing Bayesian Data Analysis 2ed, by Kruschke, in chapter 10, we get two equations (10.1, 10.2) for which no hint as to how they are obtained is given...
How does one get the second equality in the following decomposition$$p(\theta_1,\theta_2,...,m|D)=
\frac{p(D|\theta_1,\theta_2,...,m)p(\theta_1,\theta_2,...,m)}{\sum_m \int p(D|\theta_1,\theta_2,...,m)p(\theta_1,\theta_2,...,m)\ d\theta_m}=\frac{\prod_m p_m(D|\theta_m,m)p_m(\theta_m|m)p(m)}{\sum_m \int \prod_m p_m(D|\theta_m,m)p_m(\theta_m|m)p(m)\ d\theta_m}$$ ?
I would think that $p(D|\theta_1,\theta_2,...,m)= p_m(D|\theta_m,m)$? I would say there's a typo in the first equality, and that the second is wrong... But I'm the one most probably wrong.
 A: I agree that the second formula has an issue, first because the $m$ on the left of the (first) equality sign is a fixed integer, while the $m$ on the right of the (second) equality sign is a running index. (Even more incoherent is the denominator where there are two $m$ indices!) And second because indeed$$p(D|\theta_1,\theta_2,m)=p_m(D|\theta_m)$$Hence it should be that
$$p(\theta_1,\theta_2,\ldots,m|D)=\dfrac{p_m(D|\theta_m)p(m)\prod_{i\ne m}p(\theta_i|m)}{\sum_\mu p_\mu(D|\theta_\mu)p(\mu)\prod_{i\ne \mu}p(\theta_i|\mu)}$$This is a most interesting formula because it shows that a prior on $\theta_i$ must defined in all models for the joint distribution $p(\theta_1,\theta_2,\ldots,m|D)$ to make sense. In my opinion, this is not coherent with model choice: a parameter $\theta_i$ only exists when the model index is equal to $i$. To introduce the value of a model parameter within another model when this parameter has no connection with the data $D$ is not coherent. Note however that such distributions (called pseudo-priors) have been used for computational purposes, the most well-known reference being Carlin and Chib (1995) who devised an alternative to reversible jump MCMC by completing the posterior distribution into a fully joint $p(\theta_1,\theta_2,\ldots,m|D)$. But this was solely for computational reasons, there was no inferential meaning to those distributions [which could even depend on the data].
