# Heuristic Factors

*15 September 2017*

In a typical sequential Monte Carlo (SMC) setting, we target a sequence of densities (defined on a sequence of supports ) while assuming we can only evaluate their unnormalized version where the normalizing constant is .

Given an initial proposal distribution and a sequence of proposal distributions from which we can sample and whose densities we can evaluate, the SMC algorithm proceeds by the propose-weight-resample loop. The full SMC algorithm using this notation is in the first part of this note.

Since SMC is a greedy algorithm, sampling according to the original sequence of targets isn’t the best thing to do if we want to target just .
In this case we *might* want to use heuristic factors.
Heuristic factors are a sequence of positive functions which modify the sequence of normalized and unnormalized targets as follows:

- Unnormalized targets ,
- Normalized targets ,
- Normalization constants .
We require so that we preserve , and .

Recall that when using an original sequence of targets, the intermediate weights are calculated as follows:
\begin{align}
w_t = \frac{\gamma_t(x_{1:t})}{\gamma_{t - 1}(x_{1:t - 1}) q_t(x_t \given x_{1:t - 1})}.
\end{align}
If we include the heuristic factors, the weights would be calculated as follows:
\begin{align}
w_t’ &= \frac{\gamma_t’(x_{1:t})}{\gamma_{t - 1}’(x_{1:t - 1}) q_t(x_t \given x_{1:t - 1})} \\
&= \frac{\gamma_t(x_{1:t}) \cdot \prod_{\tau = 1}^t h_\tau(x_{1:\tau})}{\gamma_{t - 1}(x_{1:t - 1}) \cdot \left(\prod_{\tau = 1}^{t - 1} h_\tau(x_{1:\tau})\right) \cdot q_t(x_t \given x_{1:t - 1})} \\
&= w_t h_t(x_{1:t}).
\end{align}

## Example

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