## Functional form

$$Pr(Y_n <= y') = \Lambda(\sum_{h=1}^6 \beta_{H_{n,h}} - \sum_{a=1}^6 \beta_{A_{n,a}} - \theta_{y'})$$

$$G_k \sim \mathrm{BetaBinom}(82, \Lambda(\alpha + \delta \beta_k), \phi)$$

$$\mathbf{\beta} \sim \mathrm{N}(0, \sigma)$$

## Definitions

• Data
• $$N$$: the number of seconds of regular-season play observed across all games in the data set.
• $$K$$: the number of player-years in the data set.
• $$H$$ and $$A$$: $$N \times 6$$ matrices with elements in $$\{1, ..., K\}$$ denoting home and away players on ice.
• $$Y$$: a vector of length $$N$$ with elements equal to 1 (away shot), 2 (no shot), or 3 (home shot).
• $$G$$: a vector of length $$K$$; regular-season games played by player and year.
• Parameters
• $$\beta$$: a vector of length $$K$$; rating for each player-year.
• $$\Theta$$: a vector of length $$2$$: logit-transformed overall probabilities of away and home shots for balanced teams.
• $$\alpha$$: logit-transformed expected fraction of games (out of 82) played for an average ($$\beta = 0$$) player.
• $$\delta$$: expected increase in logit fraction of games played for a one-unit increase in $$\beta$$.
• $$\phi$$: precision parameter for beta-binomial distribution.
• $$\sigma$$: overall standard deviation of NHL player ability levels.
• Functions
• $$\Lambda(x)$$: the logistic CDF $$\frac{exp(x)}{1 + exp(x)}$$
• $$\mathrm{BetaBinom}$$: the beta-binomial distribution parameterized by size (82 games), probability, and precision

## Notes

• The null skater gets a rating $$\beta_0$$.
• i.e. unless the home team has an extra attacker at time $$n$$, $$H_{n,6} = 0$$; if they are shorthanded, $$H_{n,5} = 0$$, and so on.
• $$\beta_0$$ is not subject to the normal prior, nor are games played by the null skater used to fit the beta-binomial term.
• The rating of the null skater does not vary by year.