Probability distribution of alignments

1. Miyazawa model and Probalign model

$p^{(a)}(\theta|x,x’) = \frac{1}{Z(x,x’,T)} \exp \left(\frac{S(\theta)}{T}\right),$
where $S(\theta)$ is the score of the alignment, and $Z(x,x’,T)$ is the partition function defined as follows:
$Z(x,x’,T) = \sum_{\theta \in A(x,x’)} \exp \left(\frac{S(\theta)}{T}\right),$

2. Pair hidden Markov model

$p^{(a)}(\theta|x,x’) = \pi(s_1) ( \prod_{i=1}^{n-1}\alpha(s_i \rightarrow s_{i+1}) )( \prod_{i=1}^{n}\beta(o_i|d_i) ),$
where $\pi(.)$ is the initial probability of starting state,
$\alpha(. \rightarrow .)$ is the transition probability,
and $\beta(.|.)$ is the emission probability.