Noise Model MCMC Sampling

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Noise Model MCMC Sampling

All parameters in the noise model have full conditionals which can be sampled from. Hence for all noise parameters we employ Gibbs sampling. See (22) or (20) for an introduction to Gibbs sampling. In deriving some of the full conditional distributions, expressions of the following type are often obtained:

$\displaystyle p(\theta\vert A,B) \propto \exp\{-\frac{1}{2}[\theta^2 A - 2\theta B]\}$

(28)

where $\theta$ is the parameter whose full conditional we are deriving and

and

are functions of the other parameters in the model. This can be expressed as a Normal distribution by completing the square to give:

$\displaystyle \theta\vert A,B \sim N(B/A,1/A)$

(29)

The full conditions for the noise parameters are as follows.

Subsections

$\phi _{\epsilon _i}$
$\beta _d$ (Spatially stationary spatial model)
$\beta _{ij}$ (Spatially non-stationary spatial model) with ARD prior
$\phi _{\beta _{ij}}$ for ARD prior
$\beta _{ij}$ (Spatially non-stationary spatial model) with MRF prior
$\phi _{\beta }$ for MRF prior
$\alpha _{pi}$ with ARD prior
$\phi _{\alpha _{pi}}$ for ARD prior
$\alpha _{pi}$ with MRF prior
$\phi _{\alpha _p}$ for MRF prior