Given the following modelling specifications:
$Y_t = µ_t + σ_ee_t, \quad e_t ∼ t_1$
$µ_t = µ_{t−1} + β_{t−1} + w_t, \quad w_t ∼ N(0, σ^2_w)$
$β_t = β_{t−1} + v_t, \quad v_t ∼ N(0, σ^2_v)$
What is the conditional distribution for $Y_t|\mu_t$? I understand that for state space models, the conditional distribution of $\mu_t|Y_t$ can be approximated as follows, right?
$\mu_t|Y_t \approx Y_t|\mu_t \times \mu_t|Y_{t-1}$.
I'm unsure how and if this relates to finding an expression for $Y_t|\mu_t$ and the t-distributed errors are throwing me off. Any guidance would be appreciated.
