# Why Baum-Welch algorithm is an instantiation of EM algorithm?

$\newcommand{\E}{\mathrm{E}}$ I don't understand why Baum-Welch algorithm is an instantiation of EM algorithm.

Indeed, why computing $\alpha_t(i)$ and $\beta_t(i)$ corresponds to Expectation step. Expectation step corresponds to compute expectation over the latent variable of log-likelihood of observed variable given the parameter: $\E_{Z|X,\theta^{t-1}}[L(X|\theta)]$.

$\gamma_i(t) = E[ I(X_t=i) \vert Y, \theta] = P(X_t=i \vert Y, \theta)$
$\xi_i(t) = E[ I(X_t=i)I(X_{t+1}=j) \vert Y, \theta] = P(X_t=i, X_{t+1}=j \vert Y, \theta)$
Computing each of these conditional probabilities is the E-step, but computing them requires computing each of them requires computing all the $\alpha$'s and $\beta$'s.