Consider the observational variance discount model of Section 4.3.7. You may use the results from Problem 10.

(a) Show that the time t 1 prior ( t 1 Dt 1) G(nt 1 2dt 1 2) com bined with the beta-gamma evolution model t = t 1 t yields a conditional density p( t 1 tDt 1) that can be expressed as t 1 =

t + t 1 where

( t 1 Dt 1) G((1 )nt 1 2dt 1 2)

is independent of t

(b) Show further that p( t 1 tDT) p( t 1 tDt 1) for all T t

(c) Describe how this result can be used to recursively compute retro spective point estimates E( t DT) backward in time, beginning at t =T

(d) Describe how this result can similarly be used to recursively simulate a full trajectory of values of T T 1 1 from the retrospective smoothed posterior conditional on DT