Chapter 9 Appendix

An intuitive and less laborious way to establish properties of the HMM is to invoke properties of directed graphical models.

In a directed graphical model, the probability of a set of random variables \(\{V_1, V_2, \dots, V_n\}\) can be factored into a product of conditional probabilities, one for each parent (node) \(\text{pa} (V_i)\)

\[\begin{equation} \Pr(V_1, V_2, \dots, V_n) = \prod_{i=1}^n \Pr(V_i| \text{pa} (V_i)) \tag{9.1} \end{equation}\]

In the case of HMMs,

• \(C_1\) has no parents

• \(\text{pa} (X_k) = C_k \qquad{\text{for } k = 2, 3, \dots}\)

• \(\text{pa} (C_k) = C_{k-1} \qquad{\text{for } k = 2, 3, \dots}\)

Then the joint distribution of \(\boldsymbol{X}^{(t)}\) and \(\boldsymbol{C}^{(t)}\) is given by

\[\begin{equation} \Pr(\boldsymbol{X}^{(t)}, \boldsymbol{C}^{(t)}) = \Pr(C_1) \prod_{k=2}^t \Pr(C_k|C_{k-1}) \prod_{k=1}^t \Pr(X_k|C_k) \tag{9.2} \end{equation}\]