Model Covariates Effect on Prior Probability

Given observations $z_{1:T}$, and assume that we can divide covariates into the ones for prior ($z_t^{(pr)}$) and ones for model ($z_t^{(obs)}$) . The joint log-likelihood is

$$\begin{equation*} \log f(Y_{1:T}, S_{1:T}|\theta, z_{1:T}) = \sum_{t=1}^T \log P(S_t|\theta_{pr}, z_t^{(pr)}) + \sum_{t=1}^T\log f(Y_t|S_t, \theta_{obs}, z_t^{(obs)}) \end{equation*}$$

Generally, we can model the effects of covariates on initial probabilities as multinomial logistic regression.

$$\begin{align*} & \log \frac{P(S_t = i|\theta_{pr}, z_t^{(pr)})}{P(S_t = N|\theta_{pr}, z_t^{(pr)})} = z_t^{(pr)}\beta_i \\ & P(S_t = i|\theta_{pr}, z_t^{(pr)}) = \frac{\exp(z_t^{(pr)}\beta_i)}{\sum_{j=1}^N \exp(z_t^{(pr)}\beta_j)} \end{align*}$$

• Let the parameters for the baseline state (e.g.: State N) to be 0

This means when applying the EM algorithm, we need to find values $\theta_{pr}$ to maximize:

$$\begin{equation*} \sum_{t=1}^T \sum_{i=1}^N \gamma_t(i)\log P(S_t=i|\theta_{pr}, z_t^{(pr)}) \end{equation*}$$