Hyperdirichlet distributions for various types of informative trials including Bernoulli and multinomial
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Dimension of the distribution
Number of the winner
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
single_obs(d,n). Single multinomial trial:
players, and player
obs(x). Repeated multinomial trials:
trials, each amongst
length(x) players, with
x[i] games (which might be zero)
single_multi_restricted_obs(d,n,x). Single restricted
d players, player
conditional on the winner being one of
mult_restricted_obs(d,a,nobs). Multiple restricted
d players, conditional on winners being
a, etc. Player
nobs[i] times for 1 <= i <= d
Multiple Bernoulli trials between
single_bernoulli_obs(d,win,lose). Single Bernoulli
d players, with two teams (
lose). The winning team comprises
win, etc and the losing team comprises
bernoulli_obs(d, winners, losers) Repeated Bernoulli
d players. Here
lists of the same length; the elements are a team as in
single_bernoulli_obs() above. Thus game
losers[[i]] and, of course,
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
Robin K. S. Hankin
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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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