# Composition of probability density

I know probability distribution for parameter $\phi$. I have the empirical distribution/statistical distribution of $X$ that is dependent on parameter $\phi$ for $\phi \in [0,1]$. I assimilate this empirical distribution to the probability distribution for X.

1/ Can I do so? Are notions similar?

Then, knowing distribution for $\phi$, and 'empirical distribution' for $X$, I would like to compute the distribution for $X(\phi)$.

I first thought of using the inverse density function method. This gives a random number generator for my unknown distribution, if I can compute inverse cdf $F^{-1}$.

However, I cannot always compute $F^{-1}$. I thought then at some rejection method.

2/ I wonder if I can deal with this problem as a composition of probability density problem, and what solutions are at hand. I had a quick look and saw some optimization approaches for this class of problems (ex here).

3/ Finally, why not find the distribution of X multiplying each value of $X(\phi)$ by the probability density for this value of $\phi$? (simple product of underlying probability density by X values)

Thanks!

#

EDIT

I try to reformulate the problem in term of Bayesian statistics.

I have a prior $\phi$ with uniform distribution. I then know the distribution of X conditional to this prior for $\phi$, $P(X|\phi)$. From Bayes rule, I can deduce $P(\phi|X) = \frac{P(X|\phi)*P(\phi)}{P(X)}$.

Now, my prior is no more uniformly distributed. In other words, I have same parametric model for the distribution of X, but it is now parametrized with a non uniform RV $\theta$ following some know distribution with density $f_{\theta}$. I would like to know new $P(X|\theta)$. From Bayes rule, $P(X|\theta) = \frac{P(\theta|X)*P(X)}{P(\theta)}$.

My problem is solved if I have a relationship between $P(\theta|X)$ and $P(\phi|X)$. Should I plug in $P(\phi|X)$, which I know, in the equality $P(X|\theta) = \frac{P(\theta|X)*P(X)}{P(\theta)}$?

I hope that tentative explanations sound clearer. If not, could you guide me towards could formulation and solution?

#

EDIT - I try to clarify first sentence after Zen's comment, and to reformulate

With 'I have the empirical distribution/statistical distribution of $X$ that is dependent on parameter $\phi$ for $\phi \in [0,1]$.', I wanted to say: I know the distribution of data in a situation where some parameter, $\phi$, is uniformly distributed. I assume a model for the empirical distribution, that is parametrized with $\phi$.

Now, I am in another situation, where this parameter, which I consider a random variable, is having another distribution, with some known density $f_{\phi}$. I also assume that empirical distribution's model holds with the new underlying distribution.

Data can be produced in this model where the parameter distribution is no more uniform, but is $f_{\phi}$. I want to find the distribution for these data.

Thanks, apologize for naiv question and ackward stats lexicon.

Best regards.

-
Do you do this without any real data?? If so it sounds like you are talking about a parametric distribution for X and a uniform Bayesian prior on phi. But I am having trouble because we seem to be using very different terminology and yours is very unfamiliar to me. –  Michael Chernick Sep 7 '12 at 15:54
@Zen thanks. The OP is have a difficult time explaining his problem. I feel that if I can understand what the real question is I can solve it easily. So I tried to help him explain the problem. Based on his responses to my queries my answer is given based on my best guess about the problem. But I am still not sure I got it righr. –  Michael Chernick Sep 8 '12 at 5:02
That said, my advice is that you stick to standard statistical terminology. If we don't understand your terms, we can hardly give any useful help. For example, when you say that "I have the empirical distribution/statistical distribution of $X$ that is dependent on parameter $\phi$", it makes no sense. The empirical distribution depends only on the sample values. It can't depend on any paramenter. So, what are you really trying to say? –  Zen Sep 10 '12 at 17:19
Can you describe your problem without mentioning probability and statistics first? What do you observe? Which questions are you trying to answer? Do you want to predict some future thing based on the information contained in your sample? –  Zen Sep 10 '12 at 19:08
What does "I observe the number of events w.r.t. a parameter varying in $[0,1]$" mean? Parameters are unobservable. Things you can observe: prices of stocks, weights of things, heights of people, countings of some kind of occurence, temperatures, concentrations of chemicals, numbers of bugs in a computer program, etc. What do you observe? –  Zen Sep 10 '12 at 19:36

The problem posed in language that is familiar to me is that you want to determine the posterior distribution of phi given a prior distribution on phi that is

uniform on [0, 1].

X is distributed according to a parametric distribution say with density f$_φ$(x).

Take a sample of size n iid from f$_φ$(x).

The posterior distribution for φ is gotten by Bayes rule

g(φ|x) = c f$_φ$(x$_1$)f$_φ$(x$_2$)....f$_φ$(x$_n$) for 0<=φ<=1

g(φ|x) = 0 otherwise. c is the normalization constant that makes ∫g(φ|x)dx=1 where the integration is over the interval [0,1]. The uniform prior density for φ appears as the constant 1 when 0<=φ<=1 and is 0 otherwise. The product of the f$_φ$s is the likelihood function given X$_1$=x$_1$, X$_2$=x$_2$..., X$_n$=x$_n$.

Based on your edit it sounds like you want to infer the likelihood from the posterior and the prior. But I don't understand why you would want to do that. You are normally given data and you pick a prior and a model and then compute the posterior. But if by P(X) you mean ∫p(X|θ)p(θ)dθ then the formula you have is correct.

-
Thanks a lot for the time you spent already. I think that my problem is indeed one of baysian statistics, but not one of posterior distribution determination. I feel it is closer to computation of marginal likelihood problem, as here: en.wikipedia.org/wiki/Marginal_likelihood. I know the distribution for random variable $\phi$, and I have a iid data points $X=(x_{1},...,x_{n})$ with $x_{i}$ distribution $p(x_{i}|\phi)$. I am not sure it is clearer. I have distribution for $\phi$ already, what I need is distribution for X incorporating information on distribution for $\phi$. T –  antitrust Sep 8 '12 at 10:26
Thanks a lot, and I apologize: I am not a stat guy. –  antitrust Sep 8 '12 at 10:26
I tried to give my question a new start: I explain how I think I can use the solution you propose in your answer for my problem. You pointed out exactly the adequate notions, and I try to reformulate my question with your words. –  antitrust Sep 10 '12 at 10:51
@fonjibe I have edited my answer to address your edit. –  Michael Chernick Sep 10 '12 at 12:52
Thanks a lot. Yes, this is what I mean with $P(X)$. Maybe the naive explanation of the problem I try to give to Zen above tells more about what I am looking for. I am not given data 'in world 2', to take the same naive vocab, I am looking for data distribution in world 2 knowing the distribution 'in world 1' and assuming same 'model' in both worlds. I hope I can do some progress. –  antitrust Sep 10 '12 at 19:34