Posterior Distribution for Quantities of Interest

Let be the parameter vector. For the Cox model, the ’s are the regression coefficients ’s, and for the piecewise constant baseline hazard model, the ’s consist of the baseline hazards ’s (or log baseline hazards ’s) and the regression coefficients ’s. Let be the chain representing the posterior distribution for .

Consider a quantity of interest that can be expressed as a function of the parameter vector . You can construct the posterior distribution of by evaluating the function for each in . The posterior chain for is Summary statistics such as mean, standard deviation, percentiles, and credible intervals are used to describe the posterior distribution of .

Survival Distribution

Let be a covariate vector of interest.

Cox Model

Let be the observed data. Define

Consider the th draw of . The baseline cumulative hazard function at time is given by

For the given covariate vector , the cumulative hazard function at time is

and the survival function at time is

Piecewise Exponential Model

Let be a partition of the time axis. Consider the th draw in , where consists of and . The baseline cumulative hazard function at time is

where

For the given covariate vector , the cumulative hazard function at time is

and the survival function at time is