Re: Constrained non linear regression using ML

[Corrado]

Fredrik Lingvall wrote:
> > 3) the pdf of e is dependent on E(y)
> >     
>
> Note that y is your data and it is not distributed at all - it is the
> numbers that your data recording machine gave you. The error is just
> something you "add" because you don't have perfect knowledge of the
> physical process that you are studying. The better your knowledge is
> (better theory/better model) the less the error becomes.
>   
I do not understand what you mean by y not being distributed at all.

What I mean is:

1) our observation are y = {y_1,y_2,....,y_t}.

2) Each y_i can be considered as an extraction or realisation of a 
random variable Y.

3) This random variable Y has a distribution pdf(Y).

4) If I build the frequency distribution of y, then this frequency 
distribution may tell me something about the distribution of Y.

5) If I understand the process that generates the measurements y = 
{y_1,y_2,....,y_t}, then I may infer something about the distribution of Y.

I thought this was the approach, for example, of

Ferrari, S.L.P., and Cribari-Neto, F. (2004). 
Beta Regression for Modeling Rates and Proportions
Journal of Applied Statistics, *31*(7), 799-815.

Could you please explain me where am I going wrong?

> No I would not say that it is "an approximation by desperation". The
> Gaussian assumption is a very conservative and safe assumption.
>   
OK.
> > If I understand well, the approach you are proposing is Bayesian,
> > assuming a prior exponential positive distribution for the theta.
> >
> > I do not know the Bayesian approach, thanks a lot for the books.
> > Unfortunately, we do not have the Gregory book in our library, but I
> > will download the other one. I stuck at home with bad back and the
> > connection is not great, so as soon as I am back .... :D
> >
> > A the same time, I would like to initially use ML for comparison with
> > some previous work, before moving into Bayesian.
> >     
>
> The computational difference may not be as huge as you think. In ML you
> just try to maximize L(theta) and in MAP (Bayesian) you maximize
> p(theta|I)*L(theta) so you end up with an optimization problem in both
> cases. When you do ML you don't use all info you have (bounds of the
> parameters etc). Using such information often improve your estimates
> significantly.
>   
I do not understand: What is the advantage of applying MAP if I still 
have to calculate L(theta),
which assumes I know about the distribution of the error?

I had the impression that MAP was not completely Bayesian
and that a completely Bayesian approach would assume
only very vague prior knowledge of the distributions of the thetas.

Am I wrong?

Best,

--
Corrado Topi
PhD Researcher
Global Climate Change and Biodiversity
Area 18,Department of Biology
University of York, York, YO10 5YW, UK
Phone: + 44 (0) 1904 328645, E-mail: address@hidden