Consider a dynamic regression DLM for a univariate time series, namely yt = Ft t+ t with t N(0v) and v known. Suppose a random walk evolution for t so that G = I and t = t 1+ t and t N(0vWt)
where Wt is de ned by a single discount factor With an initial prior 0 D0 N(m0 vC0) it follows for all t 1 that t Dt N(mt vCt)
where (mtCt) are updated by the usual ltering equations.

(a) Show that the updating equations can be written in an alternative form using precision matrices as, for all t > 0 mt =Ct(R1 t mt 1+Ftyt) and C1 t =R1 t +FtFt where Rt = Ct 1+Wt

(b) Show that C1 t = tC1 0 + t r=1 t rFrFr t mt = tC1 0m0+ t r=1 t rFryr

(c) Show that C1

(d) Interpret these results in connection with the role and choice of the discount factor