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!

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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?

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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.
 A: 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.
