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Suppose at step $t$ the particle approximation of SMC in $d$ dimensions is given by $\sum_{k=1}^N w_k\delta(\vec{x}-\vec{x}_k)$, and at the subsequent step, $t+1$ (after using Bayes' law to update with a new data point), the particle approximation is given by $\sum_{k=1}^N w_k'\delta(\vec{x}-\vec{x}_k)$. Suppose also, to be concrete, that both of these distributions are good at approximating multivariate normal distributions with parameters $(\vec\mu,\Sigma)$ and $(\vec\mu',\Sigma')$, respectively.

I can think of two ways of defining the KL divergence:

  1. The probability of particle $\vec{x}_k$ is $w_k$ (or $w_k'$), so use the formula for discrete distributions, $D_{KL}=\sum_{k=1}^N w_k'\log(w_k'/w_k)$.
  2. Use the formula for KL divergence of two normal distributions, where we use estimates of the multivariate normal parameters derived from the particles (ex. $\mu=\sum_{k=1}^N w_k\vec{x}_k$): $$ D_\text{KL} = { 1 \over 2 } \left\{ \mathrm{tr} \left( \Sigma^{-1} \Sigma' \right) + \left( \mu - \mu'\right)^{\rm T} \Sigma^{-1} ( \mu - \mu' ) - d +\ln { | \Sigma | \over | \Sigma' | } \right\} $$

Would someone be kind enough to interpret both of these definitions? Moreover, in the context of adaptive experiment design, explain which one is correct (if either) and why?

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