# From evidence to belief

*Notes on probability · Equations, explained*

A model starts with a prior belief. An observation changes it. Bayes’ rule describes that update, keeping the evidence and the uncertainty in the same expression.

$$
p(\theta\mid x) = \frac{p(x\mid\theta)\,p(\theta)}{\int p(x\mid t)\,p(t)\,dt}
$$

## The shape of uncertainty

The normal density centers on $\mu$. Its spread is controlled by $\sigma$, with the normalization keeping the total area equal to one.

$$
f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)
$$

| Quantity | Definition |
| :--- | :--- |
| Mean | $\mathbb{E}[X]=\mu$ |
| Variance | $\operatorname{Var}(X)=\sigma^2$ |
| Score | $z=\frac{x-\mu}{\sigma}$ |

## Two variables, one picture

A covariance matrix holds the individual variances and the relationship between the variables.

$$
\Sigma=\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2\\\rho\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix}
$$

> Good notation lets the reader move between the idea, the equation, and the explanation without losing their place.