In RobinHankin/hyper2: The Hyperdirichlet Distribution, Mark 2

knitr::opts_chunk$set(echo = TRUE)
library(hyper2)

Partial rank observations: someone didn't win

(see also inst/loser.Rmd). Here we consider a situation in which we make an observation that is only very very weakly informative. We consider the observation ${\mathcal O}$ that a particular player does not win: the order statistic is unobserved and we know nothing about it except for the fact that (wlog) player 1 ($p_1$, aka "me") is not first placed. We consider whether there is evidence to support two different assertions:

$H_1\colon p_1=p_2=\ldots=p_n=\frac{1}{n}$
$H_2\colon\exists i\neq 1 \mbox{ with } p_1\geq p_i>0$

Our alternative would be that $(p_1,\ldots,p_n)$ is unconstrained except for the unit sum constraint:

$H_0\colon\sum p_i=1$ (that is, no restriction)

(see how $H_2$ implies that someone is worse than me). Observe that $H_1$ and $H_2$ correspond to a restriction on allowable parameter space. To understand the informativeness of ${\mathcal O}$, we will consider it in conjunction with some highly informative observations ${\mathcal I}$ which are order statistics:

(H <- 
    rank_likelihood(c(a=3,c=2,b=1,d=4)) + 
    rank_likelihood(c(b=2,c=3,d=1    )) + 
    rank_likelihood(c(d=2,a=3,c=1    )) + 
    rank_likelihood(c(b=1,d=2        )) )

We will firstly consider these observations in isolation, then later combine them with ${\mathcal O}$ which we will take to be (conditionally) independent. The first step is estimate the players' strengths:

(mH0 <- maxp(H))
loglik(indep(mH0),H)

Above, mH0 corresponds to an unconstrained optimization. Note the low but nonzero estimated strength of p_1. Observe in passing that we may reject $H_z\colon p_1=0$ unconditionally: we can deduce using elementary arguments that the likelihood of $H_z$ is zero: if $p_1=0$, then $p_4=0$ from the first line [because $p_4$ beat $p_1$ on that occasion], and then $p_1=0$ from line 2; then $p_3=0$ from line 3, $p_5=0$ from line 4 and finally $p_2=0$ from line 1. Thus we can say for sure that $H_z$ is incorrect and we may safely ignore it henceforth.

Hypothesis 1: equality

equalp.test(H)

Thus there is no evidence to suggest that the players have different strengths: we are unable to reject $H_1$, the hypothesis of equality. Hypothesis $H_2$ is rather tricky so we will use the ersatz alternative that $p_1\geqslant p_4$ (we choose $p_4$ on the grounds of its low estimated value). Then

samep.test(H,c(1,4))

There is no reason to reject our null that $p_1\geqslant p_4$, for the support falls below the two units of support criterion. Further, see how the constraint is active, for at the evaluate $p_1=p_4$ (the numerical value of small is immaterial).

Additional uninformative observation

Suppose we now make a perfectly uninformative observation: in an independent trial, the order statistic for the five competitors was one of the $5!=120$ possible orders. How does this change our relative support for $H_0$ against $H_1$ and $H_2$? It should make no difference.

library("partitions")
H # as before, here for convenience
M <- perms(5)   # uninformative!
SL <- list()
for(i in seq_len(ncol(M))){
    jj <- M[,i,drop=TRUE]
    names(jj) <- letters[seq_along(jj)]
    SL[[i]] <- H + rank_likelihood(jj)
}
SL <- as.suplist(SL)

So SL is a compound object that gives a likelihood function for the joint observation of the order statistics and the uninformative data a subsequent (independent) trial was in any of the $5!=120$ possible orders. We can find the MLE for different observations:

n <- 5 # number of players
small <- 1e-4
mSL0 <- maxplist(SL)
mSL1 <- maxplist(SL,startp=c(small*2,rep(small,n-2)),fcm=rbind(c(1,0,0,-1)),fcv=0)
mSL2 <- equalp(SL)

We use like_single_list() to calculate the likelihoods at the different evaluates; note that this function returns likelihoods, not supports.

Supp2 <-
      log(c(
      H0 = like_single_list(indep(mSL0),SL),
      H1 = like_single_list(indep(mSL1),SL),
      H2 = like_single_list(indep(mSL2),SL)))
(Supp2 <- Supp2-max(Supp2))