# bernoulli: Hyperdirichlet distributions for various types of informative... In hyperdirichlet: A Generalization of the Dirichlet Distribution

## Description

Hyperdirichlet distributions for various types of informative trials including Bernoulli and multinomial

## Usage

 ```1 2 3 4 5 6 7``` ```single_obs(d,n) obs(x) single_multi_restricted_obs(d,n,x) mult_restricted_obs(d, a, nobs) mult_bernoulli_obs(d,team1,team2,wins1,wins2) single_bernoulli_obs(d,win,lose) bernoulli_obs(d, winners, losers) ```

## Arguments

 `d` Dimension of the distribution `n` Number of the winner `x` Summary statistic `a,win,lose,winners,losers,nobs,team1,team2,wins1,wins2` Arguments as detailed below

## Details

These functions give likelihood functions for various observations. In the following, the paradigm is `d` players and the object of inference is p=(p_1...p_d) (the “skills”) with sum(p_i)=1. Different types of observation are possible.

The most informative is the unrestricted, uncensored case in which all `d` players play and the winner is identified unambiguously (`single_obs()`). However, other observations are possible, as detailed below:

• `single_obs(d,n)`. Single multinomial trial: `d` players, and player `n` wins.

• `obs(x)`. Repeated multinomial trials: `sum(x)` trials, each amongst `length(x)` players, with player `i` winning `x[i]` games (which might be zero)

• `single_multi_restricted_obs(d,n,x)`. Single restricted multinomial trial: `d` players, player `n` wins, conditional on the winner being one of `x[1]`, `x[2]`, etc

• `mult_restricted_obs(d,a,nobs)`. Multiple restricted multinomial trials: `d` players, conditional on winners being `a[1]`, `a[2]`, etc. Player `a[i]` wins `nobs[i]` times for 1 <= i <= d

• `mult_bernoulli_obs(d,team1,team2,wins1,wins2)`. Multiple Bernoulli trials between `team1` and `team2` with `team1` winning `wins1` and `team2` winning `wins2`

• `single_bernoulli_obs(d,win,lose)`. Single Bernoulli trial: `d` players, with two teams (`win` and `lose`). The winning team comprises `win[1]`, `win[2]`, etc and the losing team comprises `lose[1]`, `lose[2]`, etc.

• `bernoulli_obs(d, winners, losers)` Repeated Bernoulli trials: `d` players. Here `winners` and `losers` are lists of the same length; the elements are a team as in `single_bernoulli_obs()` above. Thus game `i` was between `winners[[i]]` and `losers[[i]]` and, of course, `winners[[i]]` won.

See examples section.

## Value

All functions documented here return a hyperdirichlet object.

## Note

The hyperdirichlet distributions returned by the functions documented here may be added (using “`+`”) to concatenate independent observations.

## Author(s)

Robin K. S. Hankin

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# Five players, some results: jj1 <- obs(1:5) # five players, player 'i' wins 'i' games. jj2 <- single_obs(5,2) # open game, p2 wins jj3 <- single_multi_restricted_obs(5,2,1:3) # match: 1,2,3; p2 wins jj4 <- mult_restricted_obs(5,1:2,c(0,4)) # match: 1,2, p1 wins 2 games, p2 wins 3 jj5 <- single_bernoulli_obs(5,1:2,3:5) # match: 1&2 vs 3&4&5; 1&2 win jj6 <- mult_bernoulli_obs(6, 1:2,c(3,5), 7,8) # match: 1&2 vs 3&5; 1&2 win 7, 3&5 win 8 jj6 <- bernoulli_obs(5,list(1:2,1:2), list(3,3:5)) # 1&2 beat 3; 1&2 beat 3&4&5 # Now imagine that jj1-jj6 are independent observations: ans <- jj1 + jj2 + jj3 + jj4 + jj5 + jj6 #posterior PDF with uniform prior likelihood ```

hyperdirichlet documentation built on May 31, 2017, 5:18 a.m.