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I am trying to develop some intuition about how weights are updated in ABC PMC. The multiple sources suggest:

$ w_t^{(i)}=\frac{\pi(x_t^{(i)})}{\sum_j^N w_{t-1}^{(j)}K_t(x_{t-1}^{(j)},x_t^{(i)})} $

where $\pi(x_t^{(i)})$ is our prior distribution evaluated at $x_t^{(i)}$ and the kernel $K$ in the denominator is "Markov kernel" or continuous "version" of transition probability matrix.

I got as far as understanding that:

  1. this is closely related to how weights are computed in importance sampling (or SIS): $ w_t(x)=\frac{\pi_t(x)}{\eta_t(x)} $
  2. Denominator is the total weighted probability of ending up at a value $x^{(i)}$ at time $t$.

What i don't understand is how our prior enters into the picture. In IS, $\pi$ would be the desired distribution known up to normalisation constant. But in ABC it is the prior which in reality could be quite far from the posterior we are trying to find, if not flat.

How does it really work? It would be nice to get an intuitive explanation.

Thanks

Vlad

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  • $\begingroup$ Have a look at Appendix A of the Toni and Stumpf paper for a derivation (n.b. you'll need to click on the pdf version to see the appendices). The intuition is roughly that every $x$ has an associated simulated dataset, and if $x$ is not close to the observations then the weight is set to zero. This explains how the likelihood enters the weights. As you point out the prior is already there. $\endgroup$ Jul 24, 2016 at 10:45
  • $\begingroup$ Thanks, for your comment. Toni and Stumpf paper is where i started. And unfortunately, while math is clear, i still miss the intuition. The question remains $\endgroup$
    – ambushed
    Oct 18, 2016 at 14:42

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