Adapted auxiliary particle filter. Recall steps (10

), (20

), and (30

) of the perfectly adapted auxiliary particle filter for model (4.33) in Section 4.3.3. Now replace the normal sampling model in (4.33) by a general sampling model f(yt

| θt).

a. Let `(θt) = ln f(yt

| θt) denote the log-likelihood factor for θt

. Find (µt

, σt)

for the quadratic approximation `(θt) ≈ −1/(2σ
2 t )(θt −µt)
2 , using a Taylor series expansion of `(θt).

b. Use this quadratic approximation to derive a variation of (4.35).

c. Using that variation of (4.35), work out appropriate variations (100), (200), and (300) of the adapted particle filter (the weights in step (300) are no longer constant).