In RobinHankin/hyper2: The Hyperdirichlet Distribution, Mark 2

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To cite the hyper2 package in publications, please use @hankin2017_rmd. Here I discuss why implementing equation 30 of [@hunter2004] for hyper2 objects representing Plackett-Luce likelihood functions is difficult. Consider the following rank observations: 724356 [that is, 7 came first, then 2, then 4 then 3 then 5 then 6] and 1372 [that is, 1 came first, then 3 then 7 then 2]. If these two observations are independent, we would have a Plackett-Luce likelihood function of

[ \left( \frac{[7]}{[234567]}\cdot\frac{[2]}{[23456]}\cdot\frac{[4]}{[3456]}\cdot\frac{[3]}{[356]}\cdot\frac{[5]}{[56]} \right)\left(\frac{[1]}{[1237]}\cdot\frac{[3]}{[237]}\cdot\frac{[7]}{[27]}\right) ]

(where $[1234]$ is short for $p_1+p_2+p_3+p_4$). Then Hunter's iteration formula gives, for example:

[ p_3\longrightarrow\frac{2}{ (p_7+p_2+p_4+p_3+p_5+p_6)^{-1}+ ( p_2+p_4+p_3+p_5+p_6)^{-1}+ ( p_4+p_3+p_5+p_6)^{-1}+ ( p_3+p_5+p_6)^{-1}+ (p_1+p_3+p_7+p_2)^{-1}+ ( p_3+p_7+p_2)^{-1} }]

or, more compactly, [p_3\longrightarrow \frac{2}{[724356]^{-1}+[24356]^{-1}+[4356]^{-1}+[356]^{-1}+[1372]^{-1}+[372]^{-1}}. ]

However, the hyper2 object corresponding to the PL likelihood function would be

[ \frac{ [1][2][3]^2[4][5][7]^2 }{ [1237]\cdot [23456]\cdot [234567]\cdot [237]\cdot [27] \cdot [3456] \cdot [356]\cdot[56] } ]

(Note that the brackets have been rearranged and also the brackets themselves. Also we have no idea which term on the numerator goes with which one on the denominator). The desired iterative step would be

$p_1\longrightarrow\frac{1}{[1237]^{-1}}$

$p_2\longrightarrow\frac{1}{[234567]^{-1}+[23456]^{-1}}$

$p_3\longrightarrow \frac{2}{[273456]^{-1}+[23456]^{-1}+[3456]^{-1}+[356]^{-1}+[1237]^{-1}+[237]^{-1}}$

$p_4 \longrightarrow\frac{1}{[724356]^{-1}+[24356]^{-1}+[4356]^{-1}}$

$p_5\longrightarrow\frac{1}{[273456]^{-1}+[23456]^{-1}+[3456]^{-1}+[356]^{-1}+[56]^{-1}}$

$p_6 \longrightarrow\frac{0}{[6]^{-1}}$

$p_7 \longrightarrow\frac{2}{[273456]^{-1}+[27]^{-1}}$

and it does not seem at all easy to convert the hyper2 object to Hunter's iteration scheme. The numerators of the fractions are just the powers of the singleton terms in the likelihood function. But the denominators are harder. To update $p_3$, for example, we would start with the denominator $[1237]\cdot[23456]\cdot[234567]\cdot [237]\cdot[27]\cdot[3456]\cdot[356]\cdot[56]$ and then extract just the brackets containing 3 to get $[1237]\cdot[23456]\cdot[234567]\cdot [237]\cdot[3456]\cdot[356]$ and then somehow identify two increasing chains

$[27]\longrightarrow[237]\longrightarrow [1237]$

and

$[356]\longrightarrow[3456]\longrightarrow[23456]\longrightarrow[234567]$