J.Q. Smith and Antonio Santos
to t. This corresponds to sampling from
f ( αt,k∣Dt ) a f ( yt ∣αt ) f ( at |at_ ɪ) ∏k, k =1 ,...,m (7)
where π k represents the weight given to each particle. The aim is then to sample first
from f (k∣Dt) and then from f (αt∣k, Dt), obtaining the sample {(αt,j, kj) ; j = 1, . . . , m}.
The marginal density f (αt∣Dt) is obtained by dropping the index k.
This resolves the problem of too many states with negligible weight being carried
forward. However, the problem of defining a good approximation to the target
distribution still remains. One of the simplest approaches is to define
g (αι,k Dt) a f yyt∖μu) f (at\at_ɪ) ∏k
(8)
where μt,k is the mean, mode or a highly probable value associated to f (at|at_ɪ).
It can easily be seen that
g (k|Dt)
a f f (yt∣μt,k) f (at\a_ɪ) πk dαt
= f {yt∖μt,k)
(9)
(10)
This density is used to define the first stage weights. These are the ones used to
sample the index that tell us which particles at t - 1 are used to define the posterior
distribution at t. Given a set of indexes, the states are drawn from f (at\at_ɪ,k) and
the second stage weights are defined as
f = f (yt∣αtj)
wj f (yt∖μtj)
(11)
The information contained in yt is carried forward through first stage weights. After
the particles at_ɪ,k, k = 1,... ,m are chosen, the densities used, f (at\at_ɪ,k), k =
1, . . . , m do not depend any further on yt .
G.E.M.F - F.E.U.C.