Hyperdirichlet distributions for various types of informative trials

Description

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

Usage

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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

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# 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

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