Review1 of Liber De Ludo Aleae (Book on Games of Chance) by ...

Ψ ( θ , θ , θ ,...) dθ dθ dθ ... × Π{
φ( x , θ , θ , θ ,...) dx }.
1 2 3 1 2 3 p 1 2 3 p
p=
1
If we integrate this over all possible values <strong>of</strong> θ1, θ2, θ3,... and
divide the original expression by the integral we shall then have
a perfectly definite value for the probability…that
θ , θ , θ ,... shall lie in any assigned limits.”
1 2 3
n
It is noted that this is a direct argument, which provides the frequency distribution <strong>of</strong> the population
parameters θ. Fisher’s caution relates to cases in which the function Ψ is not known, and is then taken to
be constant. He argues that this assumption is as arbitrary as any other, and will have inconsistent results.
While an example is not given in the Inverse Probability paper, it is helpful to consider an illustration
provided elsewhere by Fisher, related by Anders Hald in A History <strong>of</strong> Mathematical Statistics From 1750
to 1930 1 .
Consider the posterior probability element:
Then if the parameter ς is defined by
such that
θ θ ∝ θ −θ θ ≤θ ≤
a n−a P( |a,n)d (1 ) d , 0 1
1 1
sin ς = 2θ-1, - π ≤ς≤ π
2 2
ς = arcsin(2θ −1)
and ς is assumed to be uniformly distributed, the posterior probability element becomes:
Since
it follows that:
ς ς ∝ ς − ς ς
a
n−a P( |a,n)d (1+sin ) (1 sin ) d
dς darcsin(2θ
−1)
=
dθ dθ
1 d(2θ−1)
= ⋅
2
1 −(2θ−1) dθ
1 1
− −
2 2
= θ (1 −θ)
1 1
− −
2 2
dς = θ (1 −θ)
dθ,
1 A History <strong>of</strong> Mathematical Statistics From 1750 to 1930, John Wiley and Sons Inc., 1998, page 277.
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