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10 changes: 5 additions & 5 deletions dynamax/hidden_markov_model/models/abstractions.py
Original file line number Diff line number Diff line change
Expand Up @@ -526,23 +526,23 @@ class HMM(SSM):

The model is defined as follows

$$z_1 \mid u_1 \sim \mathrm{Cat}(\pi_0(u_1, \\theta_{\mathsf{init}}))$$
$$z_t \mid z_{t-1}, u_t, \\theta \sim \mathrm{Cat}(\pi(z_{t-1}, u_t, \\theta_{\mathsf{trans}}))$$
$$y_t | z_t, u_t, \\theta \sim p(y_t \mid z_t, u_t, \\theta_{\mathsf{emis}})$$
$$z_1 \mid u_1 \sim \mathrm{Cat}(\pi_0(u_1, \theta_{\mathsf{init}}))$$
$$z_t \mid z_{t-1}, u_t, \theta \sim \mathrm{Cat}(\pi(z_{t-1}, u_t, \theta_{\mathsf{trans}}))$$
$$y_t | z_t, u_t, \theta \sim p(y_t \mid z_t, u_t, \theta_{\mathsf{emis}})$$

where $z_t \in \{1,\ldots,K\}$ is a *discrete* latent state.
There are parameters for the initial distribution, the transition distribution,
and the emission distribution:

$$\\theta = (\\theta_{\mathsf{init}}, \\theta_{\mathsf{trans}}, \\theta_{\mathsf{emis}})$$
$$\theta = (\theta_{\mathsf{init}}, \theta_{\mathsf{trans}}, \theta_{\mathsf{emis}})$$

For "standard" models, we will assume the initial distribution is fixed and the transitions
follow a simple transition matrix,

$$z_1 \mid u_1 \sim \mathrm{Cat}(\pi_0)$$
$$z_t \mid z_{t-1}=k \sim \mathrm{Cat}(\pi_{z_k})$$

where $\\theta_{\mathsf{init}} = \pi_0$ and $\\theta_{\mathsf{trans}} = \{\pi_k\}_{k=1}^K$.
where $\theta_{\mathsf{init}} = \pi_0$ and $\theta_{\mathsf{trans}} = \{\pi_k\}_{k=1}^K$.

The parameters are stored in a :class:`HMMParameterSet` object.

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