$a_i$

This is sampled from using Metropolis-Hastings. Although, note that we could have used Gibbs sampling. Metropolis-Hastings requires that the following terms are updated:

$$\prod_t p(q_{it}\vert\beta,\alpha,q_{i\tilde{t}<t},\phi_{\epsilon_i})$$ (51)

$$\prod_{t,j\in{\cal N} i} p(q_{jt}\vert\beta,\alpha,q_{i\tilde{t}<t},\phi_{\epsilon_j})$$ (52)