| bernoulli {hyperdirichlet} | R Documentation |
Hyperdirichlet distributions for various types of informative trials including Bernoulli and multinomial
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
# 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