wiki/scratch.page

---
title: Scratch Page
toc: no
format: markdown
...

# Math Stuff

On observation of $\mathcal{D}$, the *likelihood* of hypothesis
$\mathcal{R}_{\alpha}$ is
$\mathit{P}(\mathcal{D}|\mathcal{R}_{\alpha})$.

## Fingerprint Variance

Additionally, we associate a collective **RSS Variance**
$\sigma_{F_s}$ with each fingerprint, which is a weighed average of
the RSS values of each of the vector elements using $C_i$ as the
weight. It is calculated in this manner:

$$
\sigma_{F_s} = \frac{\sum_{i \in F_s} \sigma_i\, C_i}
                    {\sum_{i \in F_s} C_i}
$$

# Bayesian Regression

First, specify a set of probabilistic models of the data.

Let a member of this set be denoted by $\mathcal{R}_\alpha$

$\mathcal{R}_\alpha$ has a *prior* probability $P(\mathcal{H}_\alpha)$

On observation of $\mathcal{D}$, the *likelihood* of hypothesis
$\mathcal{R}_{\alpha}$ is
$\mathit{P}(\mathcal{D}|\mathcal{R}_{\alpha})$.

The *posterior* probability of $\mathcal{R}_{\alpha}$ is then given by
$\mathit{P}(\mathcal{H}_{\alpha})\mathit{P}(\mathcal{D}|\mathcal{H}_{\alpha})$

This follows from **Bayes' Theorem** which says

$$
P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}
$$