Consider the case where we have \(y_{t}=1\) where \(C\) wins, and \(y_{t}=0\) where \(D\) wins. Recall that our model is:

\begin{equation} D \sim \text{Bern}\qty(\sigma\qty(\delta)) \end{equation}

where some \(\delta = s_{C} - s_{D}\). Now, let’s be Bayesian and compute:

\begin{equation} \delta \sim \mathcal{N}\qty(0, \tau^{2}) \end{equation}

where we say \(\delta\) starts at some measure \(\tau\).

MAP estimate

We want to find the mode of the posterior. So, we can write a MAP given this setup with Bayes rule:

\begin{equation} p\qty(\delta \mid D) = \frac{p\qty(D | \delta) p\qty(\delta)}{\int p\qty(D|d) p\qty(d) \dv{d}} \end{equation}

suppose you are using a Gaussian prior for \(\delta\) and Bernouil generator for the data, we can write:

\begin{equation} p\qty(\delta \mid D) \propto \sigma\qty(\delta)^{8} \qty [1 - \sigma\qty(\delta)]^{2} \exp \qty(-\frac{\delta^{2}}{2}) \end{equation}

for \(\tau = 1\) (or 0? surely 1). For \(\tau\):

\begin{equation} J \qty(\delta) = -w \log \sigma \qty(\delta) - \qty(n-w) \log \qty(1-\sigma\qty(\delta)) \end{equation}

where \(w\) is the number of wins and \(n\) the number of losses.

Laplace approximation

Expand the negative log posterior to the second order,