Reconstructing Evidence

Use case

Evidence is neither measured or modeled. However one can calculated using the numerator of the Bayes Theorem.

$$\boxed{C=\sum _{i\in \mathcal{S}}p(\mathbf{x}\mid s=i)\cdot P(s=i)}$$

Derivation

Using the Bayes Classifier

$$\sum _{i\in \mathcal{S}}P(s=i\mid \mathbf{x})=\sum _{i\in \mathcal{S}}\frac{p(\mathbf{x}\mid s=i)P(s=i)}{p(\mathbf{x})}=1$$

Separating $RHS$

$$1=\sum _{i\in \mathcal{S}}\frac{p(\mathbf{x}\mid s=i)P(s=i)}{p(\mathbf{x})}$$

Multiply $p(x)$ on both sides

$$\sum _{i\in \mathcal{S}}p(\mathbf{x})=\sum _{i\in \mathcal{S}}p(\mathbf{x}\mid s=i)P(s=i)$$

Evidence $C$

$$\boxed{C=\sum _{i\in \mathcal{S}}p(\mathbf{x}\mid s=i)P(s=i)}$$

Note: Here $S$ is the class, $\mathbf{x}$ is the feature vector that can be used to make the classification.