Re: Maximum likelihood estimator and multiple maxima

> Maximum likelihood is NOT the same as minimum chi-squared.

I would ask you some further explanation/pointers why not. In the standard
practice of this field it is assumed independent observations, each with
gaussian distribution. Hence the likelihood is a product of gaussians and
it becomes proportional to exp(-chi-square/2).

> The likelihood function is always equivalent to the minimal
> sufficient statistic; this does not make the location of
> its maximum necessarily the best estimator.

Which estimators could I use instead?

> are many situations where this occurs. But it does mean that
> the data cannot distinguish between these points, and only the
> prior (see Robert Dodier's response) can separate them.

I will have to look more closely at that, but it sounds to me that I would
need to give prior probability distributions on the parameters of the fit. If
so, probably I would be in trouble as the most that I could say about is
that they are within a region. Hence I would prescribe a uniform distribution
inside.

> Even if careful calculation separates them, the data do not
> treat the points as much different. In this case, it is still
> the prior, and the loss function, which are the only real guides
> you have as to which action to take. If you have widely separated
> maxima with likelihood ratios differing by a factor of two, the
> integrated areas may even reverse the situation. A risk analysis
> is clearly needed, not just a likelihood calculation.

I am not familiar with the terms "loss function" and "risk analysis". Any
pointer on these subjects would be welcomed.
.

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