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 $50\,\%$ 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 $50\,\%$ 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.