\documentclass{article}
\usepackage{amsmath, amssymb}
\title{On Bayesian updating under partial evidence}
\author{S. Adekunle}
\begin{document}
\maketitle

\section{Posterior derivation}
Let $\theta \sim \mathrm{Beta}(\alpha, \beta)$ and observe $k$ successes in $n$ trials. Then
\begin{equation}
  p(\theta \mid k, n) \propto \theta^{\alpha + k - 1}(1 - \theta)^{\beta + n - k - 1}.
\end{equation}
The posterior is $\mathrm{Beta}(\alpha + k,\ \beta + n - k)$, with mean
\[ \mathbb{E}[\theta \mid k, n] = \frac{\alpha + k}{\alpha + \beta + n}. \]

\section{Discussion}
% TODO: connect to evidence aggregation in section 3.
\end{document}