Re: Constrained non linear regression using ML

[Fredrik Lingvall]

On 03/23/10 09:10, Corrado wrote:
> 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).

No I would not say that your data is random (if this is the case then I
guess your in trouble :-)

In (Baysian) probability theory there is really no such thing as random
variables.  A probability density function (PDF) is just a measure of
your state of knowledge. If the PDF for a parameter is very spread-out
then you don't know much about your variable and if it's very sharp
(peaked) then you have a great deal of information about the parameter.

The data is the actual numbers that you got when you did the
measurements and they are not distributed. What is distributed is your
knowledge about the parameters of the model and your knowledge about the
errors (the "noise").

Phil Gregory's book explains this in a good and easy to read way :-)

>
> 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.

If you have a good understanding of the process then you should be able
to design a good (parametric) model for your problem. You then make an
inference about your model parameters not the data.

/Fredrik