way there are more intervals in areas where there is more information). The selection of the number of events per interval can be done using a model comparison measure such as the Watanabe-Akaike information criterion (WAIC) descibed in Section \[subsec:Model-comparison-and prediction\].

Prior specification\[subsec:Prior-specification\]
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In order to complete the model specification, one needs to define the prior process for the regression coefficients. One of the simplest and most applied smoothing priors on $\mathbf{\boldsymbol{\beta}}_{j}$ is the random walk

$$\mathbf{\boldsymbol{\beta}}_{j}=\mathbf{\boldsymbol{\beta}}_{j-1}+\epsilon_{j},\,\,\,\,\,\,\epsilon_{j}\sim N\left(0,\mathbf{U}_{j}\right).\label{eq:prior}$$

This process is adopted by @wagner2011bayesian, @fahrmeir1994dynamic and @hemming2002parametric, and is a special case of the more general first order random walk process for parameter evolution suggested by @gamerman1991dynamic. Clearly, if $\mathbf{U}_{j}$ is a zero matrix then there is no change in the regression coefficients and the dynamic model reduces to the standard proportional hazards model. Otherwise, $\mathbf{\boldsymbol{\beta}}_{j}$ varies over time and larger values for the entries in $\mathbf{U}_{j}$ indicate higher variation in $\mathbf{\boldsymbol{\beta}}_{j}$.

In many applications,$\mathbf{U}_{j}$ is held constant (i.e, $\mathbf{U}_{j}=\mathbf{U}$) and is assumed an unknown parameter. For a diagonal $\mathbf{U}_{j}$, both lognormal [@hemming2002parametric] and inverse gamma [@sargent1997flexible; @wagner2011bayesian] priors have been suggested for the diagonal components. For a full matrix $\mathbf{U}_{j}$, @gamerman1998markov assumes an inverse Wishart prior on $\mathbf{U}_{j}$. 