In addition, we also note some examples of possible choices of the backward kernel given along with the corresponding incremental IS weights.
- Mixture Backward kernel. Given a mixture mutation kernel in Equation (7.109), the equivalent backward kernel is given by
(7.115)with and . In this case, the incremental IS weight can be written in the following form, with respect to an index auxiliary random variable for the mixture It that was sampled:(7.116)
Approximate Optimal Backward kernel. One of the best possible approximations to the optimal backward kernel is to consider replacing vt (θt) with πt (θt), to get
which would give an incremental importance weight of(7.118)
Note that if resampling has occurred at time t — 1, then this kernel is already equivalent to the optimal choice and therefore its particle approximation is already the optimal option. In general, if using ...