hyperdirichlet (version 1.4-2)

bernoulli: Hyperdirichlet distributions for various types of informative trials

Description

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

Usage

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

Value

• All functions documented here return a hyperdirichlet object.

Details

These functions give likelihood functions for various observations. In the following, the paradigm is d players and the object of inference is $p=(p_1,\ldots,p_d)$ (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:dplayers, and playernwins.
• obs(x). Repeated multinomial trials:sum(x)trials, each amongstlength(x)players, with playeriwinningx[i]games (which might be zero)
• single_multi_restricted_obs(d,n,x). Single restricted multinomial trial:dplayers, playernwins, conditional on the winner being one ofx[1],x[2], etc
• mult_restricted_obs(d,a,nobs). Multiple restricted multinomial trials:dplayers, conditional on winners beinga[1],a[2], etc. Playera[i]winsnobs[i]times for$1\leq i\leq d$
• mult_bernoulli_obs(d,team1,team2,wins1,wins2). Multiple Bernoulli trials betweenteam1andteam2withteam1winningwins1andteam2winningwins2
• single_bernoulli_obs(d,win,lose). Single Bernoulli trial:dplayers, with two teams (winandlose). The winning team compriseswin[1],win[2], etc and the losing team compriseslose[1],lose[2], etc.
• bernoulli_obs(d, winners, losers)Repeated Bernoulli trials:dplayers. Herewinnersandlosersare lists of the same length; the elements are a team as insingle_bernoulli_obs()above. Thus gameiwas betweenwinners[[i]]andlosers[[i]]and, of course,winners[[i]]won.

See examples section.

Examples

# 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