Hyperdirichlet distributions for various types of informative trials
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 playern
wins. 
obs(x)
. Repeated multinomial trials:sum(x)
trials, each amongstlength(x)
players, with playeri
winningx[i]
games (which might be zero) 
single_multi_restricted_obs(d,n,x)
. Single restricted multinomial trial:d
players, playern
wins, conditional on the winner being one ofx[1]
,x[2]
, etc 
mult_restricted_obs(d,a,nobs)
. Multiple restricted multinomial trials:d
players, conditional on winners beinga[1]
,a[2]
, etc. Playera[i]
winsnobs[i]
times for 1 <= i <= d 
mult_bernoulli_obs(d,team1,team2,wins1,wins2)
. Multiple Bernoulli trials betweenteam1
andteam2
withteam1
winningwins1
andteam2
winningwins2

single_bernoulli_obs(d,win,lose)
. Single Bernoulli trial:d
players, with two teams (win
andlose
). 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:d
players. Herewinners
andlosers
are lists of the same length; the elements are a team as insingle_bernoulli_obs()
above. Thus gamei
was betweenwinners[[i]]
andlosers[[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 jj1jj6 are independent observations:
ans < jj1 + jj2 + jj3 + jj4 + jj5 + jj6 #posterior PDF with uniform prior likelihood
