Re: prior distributions of estimated parameters



On Fri, 8 Jul 2005 Russell.Martin@xxxxxxx wrote:

> > I am thinking of the following problem.
> > Suppose I have a nonlinear model of a curve dependent
> > on several parameters. I also have an "experimental" curve
> > available in the form of a sequence of discrete
> > points having some statistical errors. However, nothing
> > is known about the distribution of these errors, so that
> > one can only assume some "error bars" for each point.
> > Under such conditions I want to find "prior" distributions
> > of the parameters that would ensure the consistency between the model
> > and the data, meaning that the model curve passes through all the error
> > bars for the "experimental" curve.
>
> It is not clear (to me at least) that the model is capable of
> generating a curve that will pass closer than the error bars
> at all the points. Is there some reason why it should? If
> there is, how do you know that the model is not being overfit?
> Passing within all error bars is not a typical requirement,
> AFAIK. Desirable, maybe, but certainly not always the case
> even with adequate models and reasonably good data.
>
> >
> > Please note that I am NOT asking about how to perform a best
> > fit in the least squares or maximum likelihood sense,
> > which assumes (in this or other way) that the distributions
> > are normal.
>
> I do not understand the situation. Are you "assuming" error
> bars at each point or are you estimating the error bars from
> repeated measurements? If you are just "assuming", why not
> assume normal errors, at least to some level of approximation?
>

Let me reformulate the question:
Given the data error bars or a distribution of the data errors,
then what is the distribution of the parameter errors consistent with the
data.

If I say that we know the data errors, this implies that something
is known about the distribution of the data, but not necessarily
the true distribution. The error bars might well be assumed arbitrarily
based on some knowledge about the experimental setup.

I therefore used the term "prior" distribution.
Given that (limited) information, I need a method of finding the
corresponding distribution of the parameters (OK, I agree that the model
curve may not always pass through all error bars, but the fit must be
somehow consistent with the data and their errors, to be acceptable).
As the distribution of the parameters (or their errors) which I am looking
for will be concluded based on the assumed data errors (or their
assumed "prior" distribution), I think it is appropriate to call such
distribution a "prior" distribution of the parameters. If I am wrong,
please correct me.
I emphasise that in the above picture I do not assume that there
is a single "best fit" set of parameters (as is usually assumed in the
least-squares fitting). Rather, I can envisage a continuum of parameters
that ensure a consistency of the model and the data. But some of the
parameters will be more likely than the others, depending on the
information about the data distribution, and the nonlinearity of the
model.

L.B.

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