Serious post on Statistical essence.
Being frequentist and saying X ~ f(theta) is like being Bayesian and saying that you have a prior on the space of density function. But with
P(f in a particular parametric family) = 1.
This is in my opinion heavily subjective (almost shure subjective, I would say). What do you think, my friends?
I think that the statistician's work is to model data. The point is that if you choose f(x) in the most reliable way, i.e. looking at the data and not at your belief, you may think to have avoid subjectivism...
ReplyDeletesorry...avoided.
ReplyDeleteAnyhow Tony, i think "subjectivism" and not "subjectiveness".
Ti voglio bene lo stesso anche se sei bayesiano:)
Sorry Toni, but even if you are fully bayesian and you state:
ReplyDelete- X~f(\theta)
- \theta~g(\alpha),
then, again, you are saying that X is a r.v. with a density belonging to a particular parametric family: you just put some uncertainty on the parameter \theta, but still you're choosing subjectively both f and g. And even if you use a matching prior for \theta, f is subjectively choosen (and, in this case, at the end it's somehow as going back to a frequentist approach!)
I think that whenever you specify a particular shape of f and g... well actually you're doing a subjective choice.
Come on guys, it's time to move towards the amazing word of nonparametrics!
Checca
Checca, I don't want to critic others and saying that what I'm doing is better... not this time!
ReplyDeleteMine was just an open discussion.
But I know your ideas. You're right sayng that in that way you always choose arbitrarly a distribution. And if you should know what I'm doing here with Bayesian Non Parametrics...
But this is another story... keep on permuting checca...
I'm looking forward to see your work, Toni. Even more now that you're telling me that it's about bayesian nonparametrics!
ReplyDeleteI took your post as an open discussion, I like open discussions! And my last sentence was just an innocent suggestion.
Checca
Tony, I'm really interested (I don't belive me) in Bayesian Non Parametrics, but I would like to understand if it is possible to use a totally nonparametric prior (I mean, I need only its first two moments) more than use a multinomial prior with infinite parameters like you explain me... do you know something about the former?
ReplyDeleteActually you don't assume a multinomial prior. The discussion should be more deep. I don't understand properly your question... do you want to assume a non parametric prior distribution on what?
ReplyDeleteThe cool thing of BNP is that you put prior not on the parameter space as usually intent, but on the space of densities function defined with respect to some measure.
A random experiment should be: take randomly a density from the space of densities. Take a sample x from that density. Clearly as P(X=x_0) = 0 for every continuous density also P(f is Normal(or every parametric family)) = 0. This mean that if you start from the second step as usual (assume X ~ f) you are conditioning your inference on an event almost shurely impossible! That's really cool, for me...however ze un casin!
@checca
ReplyDelete...e comunque permutation test \subset Non Parametric Statistic.
I'm puzzled... I'm not be able to imagine the space of densities, or, better, in which sigma-algebra you work.
ReplyDeleteAnyway, if you don't put a prior on the parameter space, your method is useless for my idea...
Now I finally understand what Nonparametric Bayesian is.
ReplyDeleteI think it is a very nice approach, but as always I am a truly applied statistician (even when we are talking about philosophy...).
My point is: a statistical model is not the truth and (in my opinion) you should prefer a *simple* model rather than a *perfect* model. Your aim should be to find a model that works fine and that is not too far from the data.
I think that if you seek for the most "real" model, your risk is to achieve a model too complicated to handle, and eventually you will find yourself with a perfect model and some brutal (numeric) approximation.
So why don't you start with a (not numerical but substantial) approximation from the beginning?
When you choose the Gaussian model, you are saying: "I know that it is wrong, but since every model is wrong I take the simplest to handle".
This is how I see frequentist approach.
I like very much Maestro's pragmatic approach. And I'm not a crusader of Bayesian Faith, I still like likelihood very much. So:
ReplyDelete@Maestro: if you like simple methods that work, that's not only frequentist. I'm a fan of the t-test (ok, checca for small sample size it can be misleading...) but there are also simple (and without MCMC!) bayesian tools. In addition Bayesian thinking is (not for us, but for a perfect idiot like a doctor) more intuitive! Think to confidence interval and credible interval... how many of our undergrad still say "a confidence interval of level alpha contains the parameter with probability 1-alpha"...
According to me one should be Frequentist or Bayesian according to wich solution to a specific problem is clearer, more reliable and simple.
@Riccardo: your parameter is a whole density function. and you put a prior on the parameter space. but I agree... it's not easy to imagine it! It's not the usual bell shape density over a plot with theta on the x-axis...
Vago farme do spaghetti!