[Work Log] Post-CVPR-deadline; 2-part likelihood efficiency, 2-pass sampling

November 04, 2013

First day after CVPR deadline.

Comp exam - committee, planning.

Computer issues - upgrade, crashing, keyboard?

2-Pass sampling.

Evaluating the second likelihood term is still very slow, even after a 100x speedup.

First do MH using single-term likelihood. Then treat that as the proposal for the two-term likelihood. Since the first step satisfies detailed balance, we have:

\[ \begin{align} \hat p(\theta) q(\theta' | \theta) &= \hat p(\theta') q(\theta | \theta') \\ \frac{\hat p(\theta)}{\hat p(\theta')} &= \frac{q(\theta | \theta')}{q(\theta' | \theta)} \end{align} \]

Where \(\hat p(\theta)\) is the surrogate posterior, using only the single-term likelihood. Substituting this identity into the full MCMC acceptance term, we get:

\[ \begin{align} \alpha &= \left \{ \frac{p(\theta') q(\theta | \theta')}{p(\theta)q(\theta' | \theta)} \right \} \\ &= \left \{ \frac{p(\theta') \hat p(\theta)}{p(\theta)\hat p(\theta')} \right \} \\ &= \frac{L_2(\theta')}{L_2(\theta)} \end{align} \]

This obviously isn't applicable for traditional gibbs sampling, but gibbs could be used for proposal, and MH used to accept/reject

Posted by Kyle Simek
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