Bayes rule and terms with expectation

I am reading the following paper in economics; link On page 495, authors give an expression with Bayes rule. As an example, say that there is a random variable $\beta$ which can be either $\beta_L$ or $\beta_H$ . So, there is an information that you receive, say $T$ such that $$H_t =\beta X_t+\epsilon_t$$ Let the probability of having a $\beta_H$ is $\pi_t$ . Then, with the Bayes rule, we can have $$\pi_{t+1}=\frac{\pi_{t}exp\left(-p_{\varepsilon}\left(H_{t}-\beta_{H}X_{t}\right)^{2}\right)}{\pi_{t}exp\left(-p_{\varepsilon}\left(H_{t}-\beta_{H}X_{t}\right)^{2}\right)+\left(1-\pi_{t}\right)exp\left(-p_{\varepsilon}\left(H_{t}-\beta_{L}X_{t}\right)^{2}\right)}$$ My question is; is it possible to take the expectation of the probability $\pi$ . Normally, is not that $\pi_{t+1}$ is the conditional expectation of $\pi$ on the signal $H$ ? I ask this question because authors of the paper take the expectation of this probability on page 516. Also, for Bayesian updating (as in Kalman filter), the updated mean (posterior) is the expectation of the random variable, conditional on the signal. Thanks a lot in advance for any hints/suggestions.