9.6 Appendix

9.6.1 Proof of Proposition 9.1 in Section 9.5.2

To show that the sample given by (9.111) and (9.112) is a properly weighted sample with respect to p ( S _t Y _t ), we need to verify that (9.112) gives the correct weight. Assume that at time ( t -1), we have a properly weighted sample with respect to p ( S _{t -1} Y _{t -1} ). That is, assume that is drawn from some trial distribution q ( S _{t -1} Y _{t -1} ) and that the importance weight is given by , with

Equation 9.144

By (9.109) and (9.111), is drawn from the distribution . Hence, the sampling distribution for is given by . Since the target distribution is p ( S _t Y _t ), the weight function at time t is then given by

Equation 9.145

Hence, . This verifies that (9.112) gives the correct importance weight at time t.