8.2 Let S Nn(????, ) be a multivariate normal latent signal, and, given S, consider independent...
Question:
8.2 Let S ∼ Nn(????, Σ) be a multivariate normal latent “signal,” and, given S, consider independent Poisson distributed observations Zi
|S = s ∼ P(????i
), where log ????i = si
, i = 1, ..., n. Show that the first two moments of the observations and signal are given by E(Zi
) = exp (
????i +
1 2
????ii)
= gi
,say, E(Z2 i ) = g2 i exp (
????ii)
+ gi
, var(Zi
) = g2 i
{
exp (
????ii)
− 1
}
+ gi
, E(Zi Zj
) = gi gj exp (
????ij)
, cov(Zi
, Zj
) = gi gj
{
exp (
????ij)
− 1
}
, i ≠ j, E(Si
) = ????i
, var(Si
) = ????ii, E(Si Zj
) = (
????i + ????ij)
gi
, cov(Si
, Zj
) = ????ijgj
, where i, j = 1, ..., n. Note that the formulas for E(Zi Zj
) and cov(Zi
, Zj
) are valid only for i ≠ j; the formula for var(zi
) includes a nugget-like term gi not present for the covariances. On the other hand, the formulas for E(Si Zj
) and cov(Si
, Zj
) are valid for all i, j = 1, ..., n.
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