The joint density function
is mainly calculated from the edge density function and hidden variables.
Note that (9) boils down to finding the joint density function
of p(x | [Z.sup.k.sub.1]) and p(x | [Z.sup.k.sub.2]).
A variant of EM called stochastic EM uses a single realization of the joint density function p([x.sub.1:T],[y.sub.1:T]|[y.sub.1:T], [theta]) to approximate the above integral (Celeux and Diebolt, 1985)
Note that the dimensionality of p([x.sub.t]|[y.sub.1:T], [theta]) is much smaller than the dimensionality of the joint density function of the states.
The joint density function between [x.sub.t] and [x.sub.t+1] can be obtained by using (27)
If we decompose the vector [x.sub.t] into [m.sub.t] (the real money demand) and [z.sub.t] (= [RY.sub.t], [INF.sub.t], [R.sub.t])' represents determinants of money demand, then the joint density function (Equation 3) can be factorised into the conditional density function of [m.sub.t] given [z.sub.t] (i.e., [F.sub.m/zt] (m/[z.sub.t]; [[lambda].sub.1]) and the marginal density function of [z.sub.t] (i.e., [F.sub.zt] ([z.sub.t]; [[lambda].sub.2]).
In this case the factorisation of joint density function into conditional function and marginal model helps to isolate effects of these shocks.
We will derive the joint density function of the duration (that is, the time required to find a job) and the number of applications (in each period) as follows.
Then, we have for the joint density function of a completed spell of duration T (that is, one successful application at t = T), [k.sub.j] unsuccessful applications at t = 28j days for j = 1, .
ML methods are not confined to the classical case; other density functions such as the conditional joint density function
of the states and measurements, given the model parameters, can also be used (Maybeck, 1982).