where P denotes the matrix of transition probabilities. This equation can be rearranged to
(I-P)p(S0|X) =0,
(4.12)
with I being a (k × k) identity matrix. We know that by construction the k probabilities in
the vector of p(S0∣X) add up to one. Thus, with ι = (1,..., 1)' we may express this fact in vector
notation as
ιp(So∣X ) = 1.
(4.13)
In matrix notation (4.12) and (4.13) can be rewritten as
I-P
ι
p(S0|X) =
(4.14)
≡M
We premultiply this expression by (M'M) 1M' and obtain for the initial probabilities
p(So∣X ) = (M 'M )-1M '
(4.15)
4.1.4 Generating the Parameters
After having generated S , we are now able to formulate the conditional density of the parameters,
which is generally given by
p(λj∣S,λ-j,X) ∖ L(X∣S,λ) ∙p(S∣λ) ∙p(λj), (4.16)
where λ-j denotes the set of all parameters except for λj .
17
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