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

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 
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.
All functions documented here return a hyperdirichlet object.
The hyperdirichlet distributions returned by the functions
documented here may be added (using “+
”) to concatenate
independent observations.
Robin K. S. Hankin
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

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