Consider the model yt = xt + vt, where vt is Gaussian white noise with variance 2 v, xt are independent Gaussian random variables with

Consider the model yt = xt + vt, where vt is Gaussian white noise with variance σ2 v, xt are independent Gaussian random variables with mean zero and var(xt) = rtσ2 x with xt independent of vt, and r1, . . . , rn are known constants. Show that applying the EM algorithm to the problem of estimating σ2 x and σ2 v leads to updates (represented by hats)

σb2 x = 1 n

Xn t=1

σ2 t + µ2 t

rt and σb2 v = 1 n

Xn t=1

[(yt − µt)

2 + σ2 t ], where, based on the current estimates (represented by tildes),

µt = rtσe2 x

rtσe2 x + σe2 v

yt and σ2 t = rtσe2 xσe2 v

rtσe2 x + σe2 v

.