Alternatively, @west1985dynamic suggests a discounting procedure for approximating $\mathbf{U}_{j}$ in terms of a discount parameter $0<\phi<1$ that controls the amount of information transferred through intervals. Given the posterior variance $\boldsymbol{\Sigma}_{j-1}$ of the regression coefficients in the interval $I_{j-1}$, the discount factor approach approximates $\mathbf{U}_{j}=(\phi^{-1}-1)\boldsymbol{\Sigma}_{j-1}$. This allows the variance $\mathbf{U}_{j}$ to vary over time, which improves the capacity of the random walk process to adapt locally. It is also possible to assume a time-varying discount factor (see @das2013dynamic). However, in this paper, $\mathbf{U}_{j}$ is allowed to be a full matrix, $\phi$ is held constant and inference on $\phi$ is based on the WAIC described in Section \[subsec:Model-comparison-and prediction\].

Inference
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This section describes the posterior distribution and the sequential Monte Carlo sampling procedure used to sample from the target posterior distribution, $$p\left(\boldsymbol{\beta}_{1:J}|\mathbf{t}_{1:J}\right)\propto p\left(\mathbf{t}_{1}|\boldsymbol{\beta}_{1}\right)p\left(\boldsymbol{\beta}_{1}\right)\prod_{j=2}^{J}L_{j}\left(\mathbf{t}_{j}|\boldsymbol{\beta}_{j}\right)p\left(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1}\right),\label{eq:joint posterior}$$ where $p(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1})$ is defined by the expression (\[eq:prior\]). MCMC methods have been used in the literature to sample from (\[eq:joint posterior\]). @hemming2002parametric reparametrize $\boldsymbol{\beta}_{j}$ in terms of the evolution noise $\epsilon_{j}$ in (\[eq:prior\]), and apply a Gibbs sampler with a random walk Metropolis Hastings step for each $\boldsymbol{\beta}_{j}$. @wagner2011bayesian designs a Gibbs sampler where the full path $\boldsymbol{\beta}_{1:J}$ is sampled in one move using the forward filtering 