Bayesian Theorem

Bayesian Theorem and Model Evidence

With measurement $\mathbf{D}$ and model parameters $\mathbf{\theta}$

$$ Pr(\mathbf{D}|\mathbf{\theta})\times Pr(\mathbf{\theta})=Pr(\mathbf{D})\times Pr(\mathbf{\theta}|\mathbf{D}) $$ $$ {\rm Likelihood}\times {\rm Prior}={\rm Evidence}\times {\rm Posterior} $$ $$ \mathcal{L}(\theta)\times \pi(\mathbf{\theta})d\mathbf{\theta}=Z\times P(\theta)d\mathbf{\theta} $$ $$ \int \mathcal{L}(\theta)\times \pi(\mathbf{\theta})d\mathbf{\theta}=Z\times \int P(\theta)d\mathbf{\theta} $$

$$\because P(\theta)d\theta\equiv 1$$ $$\therefore Z=\int \mathcal{L}(\theta)\times \pi(\mathbf{\theta})d\mathbf{\theta}$$