Bayesian Theorem

Bayesian Theorem and Model Evidence

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

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

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