David A. Heiser wrote:
> I am going to reference Fisher as his views later on in life in the 1973 3rd
> edition of "Statistical
> Methods and Scientific Inference"
>
> "The Mathematical Likelihood assignable to every value of the unknown
> parameter p supplies a zoning of the admissible range of values more
> appropriate to the observations than by the system of cofidence belts....It
> is, like Mathematical Probability, a well-defined quantitative feature of
> the logical situations....but it is a quantity of a different kind from
> probability, and does not obey the laws of probability". (page 72)
>
> Therefore L(H|D) which is a probability, obeying probability laws, differs
> from Fisher's concept.
>
But surely L(H|D) integrated over the range of H is not necessarily equal to
unity. So L(H|D) cannot be a probability density function. [Assuming the
usual practice of defining L(H|D) as an arbitary multiple of P(D|H). The
problem still exists iif we define L(H|D) = P(D|H); as we are really saying
L(H|D)dH=P(D|H)dP and the dH & dP come back to haunt us. ]
OTOH you cannot just `normalise' L(H|D) by the value of its integral. To
be sure this will make the integral unity. But transformations -- even
monotonic ones -- of the parameter space) will give a different function.
There is little reason to believe that -- when used with an arbitary
parameterisation of parameter-space -- such a normalised L(H|D)
assigns equal values to equally plausible ranges in H-space.
Of course this is why we use maximum likehood (the position of the
maxima is invariant under monotonic transformation of the parameter
space).
OTOH I believe that the search for the `right' parameter transformations
is the underlying problem solved by `uninformative' Bayesian priors.
Peter
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