How to compute the log likelihood of a Poisson Hidden Markov Model once I know all the parameters

Question

I know that the answer is maybe pretty straightforward, but the Expectation Maximization algorithm confused me a little.

So basically my problem is this:

I have a set of data and applied the Expectation Maximization algorithm to find the transition probability and the lambdas of my Poisson Hidden Markov Model, now how do I compute the log-likelihood?

Thank you so much!

Answer 1

If you have implemented EM algorithm. Surely you will have $\alpha$ and $\beta$ values from forward backward algorithms. Here, $\alpha_t(j) = P(x_t=j,Z_{\{1:t\}}\vert \theta)$.

$Z_{\{1:t\}}$ is all the emissions (observations) up to time $t$.

$\theta$ are the learned parameters from EM.

Now the log likelihood is log($P(Z_{\{1:t\}}\vert \theta)) = log(\sum_{j} \alpha_t(j))$