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##

The dog, the hunter and the biased Bayesian
estimators

One of the most important tests to judge
the quality of an estimator,
is whether or not it is
*correct* (not biased).
Maximum likelihood estimators are
usually correct, while Bayesian estimators -- analysed within
the maximum likelihood framework -- often are not.
This could be considered a weak point; however the
Bayes estimators are simply
naturally consistent with the state
of information before new data
become available.
In the maximum
likelihood method, on the other hand, it is not clear what
the assumptions are.
Let us take an example which shows the logic of frequentistic
inference and why the use of reasonable prior distributions
yields results which
that frame classifies as distorted.
Imagine meeting a hunting dog in the country. Let us assume we
know that there is a probability
of finding the dog within a radius of 100 m centred
on the position of the hunter (this is our likelihood).
Where is the hunter? He is with probability
within a radius of 100 m around the position of the dog,
with equal probability in all directions. ``Obvious''.
This is exactly the
logic scheme used in the frequentistic approach to
build confidence regions from the estimator (the dog in this
example). This however assumes that the hunter can be anywhere
in the country. But now let us change the state of information:
``the dog is by a river''; ``the dog has collected a duck and
runs in a certain direction''; ``the dog is sleeping'';
``the dog is in a field surrounded by a fence through which he
can pass without problems, but the hunter cannot''. Given
any new condition the conclusion changes.
Some of the new conditions change our likelihood, but
some others only influence the initial distribution.
For example, the case of the dog in an enclosure
inaccessible to the hunter is exactly the problem encountered
when measuring a quantity close to the edge of its physical region,
which is quite common in frontier research.

** Next:** Choice of the initial
** Up:** Statistical inference
** Previous:** Bayesian inference and maximum
** Contents**
Giulio D'Agostini
2003-05-15