I have a fantasizing about a model here, so please keep in mind that this is not even half-baked:
I have categorical time series data $y_t\sim\text{Cat}(y\ |\ \lambda_t)$ with a hidden variable $\lambda_t\sim\text{Dir}(\lambda\ |\ x_t,\lambda_{t-1})$ which is time-dependent (dependent on previous values) and dependent on features, $x_t\in\mathbb{R}^d$.
One could specify more precisely, $p(\lambda_t)=\text{Dir}(\lambda\ |\ \alpha_t)$ where $\alpha_t=\gamma\alpha_{t-1}+(1-\gamma)\theta^Tx_t$
What do I have in mind with this model: You have a Dirichlet posterior density on a simplex where the prior to that density is the density of the previous step (like a Kalman filter). But I also have observed variables that can push that density away (this is where I start fantasizing because this is not a likelihood term like in a Kalman filter). The actual likelihood comes from the observed categorical variables $y_t$
Is something like this possible or has work on this been done?
