J.Q. Smith and Antonio Santos
fined through the second order Taylor approximation of log f (yt∣αt) around μt,k,
log g (yt αt∙μt,k) x l (μt,k) + l0 (μt,k) (α- μt,k) +1 l00 (μt,k) (α- μt,k¢2 (24)
= const - αt - 2 yt (---γ ( (αt - μt,k) + (αt 2μt,k) - 1! (25)
2 2β exp μttyk) , 2 y
Using this second order approximation, the density g (αt, k|Dt) is factorized as
in equation (15) and the factors are
g (at∖at-I,k,yt,μt,k} = n μt,k,σlk} (26)
and
where
and
g yyt∖μt,k)
exp (2 ɑ + 2β 2 exp ( μt,k )
Л * 2
l μt,k -
× exp
-_ yt (1 + μt,k)
∖ 2β2 exp (μt,k)
1j+ + yt ʌ 1 ï yt (1 + μt,k ) + μt,k _ 1
σηση 2β2 exp (μt,k)J y 2β2 exp (μt,k) ^1 2
2 β 2 σ П
2β2 + exp (-μt,k) σ2У2
(27)
(28)
(29)
(30)
As we sample from g (αt, k|Dt), an approximating sample, the elements in it
must be resampled in order to obtain a sample that gives a better approximation of
the target density f (αt, k|Dt). The weights used in this resampling step are
log wj
πj
У2
2β2 exp (αt,j )
yt (i_ αt,j (1 _ αj+μt,k ) + ( μt,k+μ2k ´´
2β2 exp μttjβ)
wj
p m=ι wi,
j = 1, . . . ,m
These are the so-called second stage weights that allow the modification of the
approximating distribution towards the target distribution. Obviously, these weights
G.E.M.F - F.E.U.C.
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