Maximum-Likelihood Reversible Transition Matrix¶

Here, we sketch out the objective function and gradient used to find the maximum likelihood reversible count matrix.

Let \(C_{ij}\) be the matrix of observed counts. \(C\) must be strongly connected for this approach to work! Below, \(f\) is the log likelihood of the observed counts.

\[f = \sum_{ij} C_{ij} \log T_{ij}\]

Let \(T_{ij} = \frac{X_{ij}}{\sum_j X_{ij}}\), \(X_{ij} = \exp(u_{ij})\), \(q_i = \sum_j \exp(u_{ij})\)

Here, \(u_{ij}\) is the log-space representation of \(X_{ij}\). It follows that \(T_{ij} = \exp(u_{ij}) \frac{1}{q_i}\), so \(\log(T_{ij}) = u_{ij} - \log(q_{i})\)

\[f = \sum_{ij} C_{ij} u_{ij} - \sum_{ij} C_{ij} \log q_i\]

Let \(N_i = \sum_j C_{ij}\)

\[f = \sum_{ij} C_{ij} u_{ij} - \sum_{i} N_i \log q_i\]

Let \(u_{ij} = u_{ji}\) for \(i > j\), eliminating terms with \(i>j\).

Let \(S_{ij} = C_{ij} + C_{ji}\)

\[f = \sum_{i \le j} S_{ij} u_{ij} - \frac{1}{2} \sum_i S_{ii} u_{ii} - \sum_i N_i \log q_i\]

\[\frac{df}{du_{ab}} = S_{ab} - \frac{1}{2} S_{ab} \delta_{ab} - \sum_i \frac{N_i}{q_i} \frac{dq_i}{du_{ab}}\]

\[\frac{dq_i}{du_{ab}} = \exp(u_{ab}) [\delta_{ai} + \delta_{bi} - \delta_{ab} \delta_{ia}]\]

Let \(v_i = \frac{N_i}{q_i}\)

\[\sum_i V_i \frac{dq_i}{du_{ab}} = \exp(u_{ab}) (v_a + v_b - v_a \delta_{ab})\]

Thus,

\[\frac{df}{du_{ab}} = S_{ab} - \frac{1}{2} S_{ab} \delta_{ab} - \exp(u_{ab}) (v_a + v_b - v_a \delta_{ab})\]