11 Hazard Smoothing

The Euler-multinomial distribution is a good place to get a little background on the hazard smoothing approach.

\[ p_j = \left(1-\exp\left(-\sum_i{r_i dt}\right)\right) \frac{r_j}{\sum_i r_i}. \] The first piece is the probability of leaving the box at an exponential rate, and the second piece is that fraction of flows going to a particular box.

Define the sums of the rows of the rate matrix. \[ r_i = \sum_{j=1}^n M_{ij} \] Define the elements of a vector of exponentiated row sums \[ \rho_i = \exp(-r_i) \]

Define the elements of a normalized state vector. \[ \tilde{s}_i = \begin{cases} 0 & r_i = 0 \\ \frac{s_i}{r_i} & \text{otherwise} \end{cases} \] With these definitions we can define the modified flow matrix. \[ F_{ij} = \begin{cases} M_{ij}\tilde{s}_i(1-\rho_i) & i \ne j \\ 0 & \text{otherwise} \end{cases} \] This modified flow matrix can now be used in the same way as the unmodified flow matrix to produce state variable updates following spec version 0.0.2.