# 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,

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