Hidden Markov Models
An elementary first-order Hidden Markov model is a probabilistic model over \(N\) observations, \(y_n\), and \(N\) hidden states, \(x_n\), which can be fully defined by the conditional distributions \(p(y_n \mid x_n, \phi)\) and \(p(x_n \mid x_{n - 1}, \phi)\). Here we make the dependency on additional model parameters, \(\phi\), explicit. When \(x\) is continuous, the user can explicitly encode these distributions in Stan and use Markov chain Monte Carlo to integrate \(x\) out.
When each state \(x\) takes a value over a discrete and finite set, say \(\{1, 2, ..., K\}\), we can take advantage of the dependency structure to marginalize \(x\) and compute \(p(y \mid \phi)\). We start by defining the conditional observational distribution, stored in a \(K \times N\) matrix \(\omega\) with \[ \omega_{kn} = p(y_n \mid x_n = k, \phi). \] Next, we introduce the \(K \times K\) transition matrix, \(\Gamma\), with \[ \Gamma_{ij} = p(x_n = j \mid x_{n - 1} = i, \phi). \] Each row defines a probability distribution and must therefore be a simplex (i.e. its components must add to 1). Currently, Stan only supports stationary transitions where a single transition matrix is used for all transitions. Finally we define the initial state \(K\)-vector \(\rho\), with \[ \rho_k = p(x_0 = k \mid \phi). \]
The Stan functions that support this type of model are special in that the user does not explicitly pass \(y\) and \(\phi\) as arguments. Instead, the user passes \(\log \omega\), \(\Gamma\), and \(\rho\), which in turn depend on \(y\) and \(\phi\).
Stan functions
real hmm_marginal(matrix log_omega, matrix Gamma, vector rho)
Returns the log probability density of \(y\), with \(x_n\) integrated out at each iteration.
The arguments represent (1) the log density of each output, (2) the transition matrix, and (3) the initial state vector.
log_omega: \(\log \omega_{kn} = \log p(y_n \mid x_n = k, \phi)\), log density of each output,Gamma: \(\Gamma_{ij} = p(x_n = j | x_{n - 1} = i, \phi)\), the transition matrix,rho: \(\rho_k = p(x_0 = k \mid \phi)\), the initial state probability.
array[] int hmm_latent_rng(matrix log_omega, matrix Gamma, vector rho)
Returns a length \(N\) array of integers over \(\{1, ..., K\}\), sampled from the joint posterior distribution of the hidden states, \(p(x \mid \phi, y)\). May be only used in transformed data and generated quantities.
matrix hmm_hidden_state_prob(matrix log_omega, matrix Gamma, vector rho)
Returns the matrix of marginal posterior probabilities of each hidden state value. This will be a \(K \times N\) matrix. The \(n^\mathrm{th}\) column is a simplex of probabilities for the \(n^\mathrm{th}\) variable. Moreover, let \(A\) be the output. Then \(A_{ij} = p(x_j = i \mid \phi, y)\). This function may only be used in transformed data and generated quantities.