Posterior Probability Distribution

The posterior probability distribution over the parameters in the null model (tables 5.6, 5.7) $M_N$ is given by Bayes' theorem [89]:

$$P(\mathbf{Q}\vert D, M_N) = \frac{P(D\vert\mathbf{Q})P(\mathbf{Q}\vert M_N)}{P(D\vert M_N)}\textrm{,}$$ (5.43)

where

$$P(D\vert M_N) = \int_{\textrm{All }\mathbf{Q}\textrm{ space... ...ert M_N)P(D\vert\mathbf{Q})\mathrm{d}^{47}\mathbf{Q}\textrm{.}$$ (5.44)

Similarly, for the main model,

$$P(\mathbf{Q}\vert D, M_M) = \frac{P(D\vert\mathbf{Q})P(\mathbf{Q}\vert M_M)}{P(D\vert M_M)}\textrm{,}$$ (5.45)

where

$$P(D\vert M_M) = \int_{\textrm{All }\mathbf{Q}\textrm{ space... ...ert M_M)P(D\vert\mathbf{Q})\mathrm{d}^{47}\mathbf{Q}\textrm{.}$$ (5.46)

The posterior probabilities of the models are

$$P(M_N\vert D) = \frac{P(D\vert M_N)P(M_N)}{P(D)}$$ (5.47)

and

$$P(M_M\vert D) = \frac{P(D\vert M_M)P(M_M)}{P(D)}\textrm{,}$$ (5.48)

where

$$P(D) = P(M_N)P(D\vert M_N)+P(M_M)P(D\vert M_M)\textrm{.}$$ (5.49)