MCMC sampling with a probability density function that have potential negative values My question might be quite strange, but I will expose you the complete issue in order for you to help me.
I am in the context of a parallel randomized clinical trial which aim is to compare two treatment options (a reference one denoted $T=0$, and an innovative one denoted $T=1$). Let's also define a marker that is measured before any treatment allocation $X$. The context of clinical trial leads to what I call the randomization constraint that states:
\begin{equation}
P(X \leq c|T=0)=P(X\leq c|T=1) \ \forall c
\end{equation}
This equation may be expressed as:
\begin{equation}
P(X \leq c|T=0,E=1)P(E=1|T=0)+P(X \leq c|T=0,E=0)P(E=0|T=0)=P(X \leq c|T=1,E=1)P(E=1|T=1)+P(X \leq c|T=1,E=0)P(E=0|T=1) 
\end{equation}
For simplicity, I will note that $F_{tk}(c)=P(X \leq c|T=t, E=k)$ the cumulative distribution function of the marker $X$ in the group $tk$, and $\rho_t=P(E=1|T=t)$ so that the previous equation is expressed as:
\begin{equation}
F_{01}(c)\times \rho_0+F_{00}(c)\times (1-\rho_0)=F_{11}(c)\times \rho_1+F_{10}(c)\times (1-\rho_1)
\end{equation}
So using this randomization constraint, it is possible to express one of the four CDF as a function of the three other ones, for example, let's express $F_{01}(c)$ as a function of the three other CDF, $\rho_0$, and $\rho_1$:
\begin{equation}
F_{01}(c) = [F_{11}(c)\times \rho_1+F_{10}(c)\times (1-\rho_1)-F_{00}(c)\times (1-\rho_0)]/ \rho_0
\end{equation}
Note: The same equation is used to obtain $f_{01}(.)$ the probability density function
Well, I need to estimate a function, let's call it $U(.)$, that depends on $F_{01}(.)$, $F_{00}(.)$, $F_{10}(.)$, and $F_{11}(.)$ and whose expression relies on the randomization constraint. I was thinking of a way to include this constraint in the estimation method of $U(.)$, so I wanted to express the $F_{01}(.)$ as a function of the three other CDF as I presented it above.
The estimation process would use the Bayesian inference by sampling from the posterior distribution of $\rho_0$, $\rho_1$, and the parameters of $F_{00}(.)$, $F_{10}(.)$, and $F_{11}(.)$.
The issue is that, for some combination of sampled parameters in the MCMC, $f_{01}(.)$ (the probability density function) might not be well-defined and could lead to a function that accepts negative values (which is quite embarassing when dealing with a probability density function), so that my estimation of $U(.)$ is not good.
What could be the solutions ? 
For example I was thinking of a way to exclude the parameter combinations that lead to bad-defined $f_{01}(.)$. But I am not sure this is a correct solution.
By advance thank you for your help, I hope my problem is well-exposed here. If you need more details do not hesitate to ask.
 A: The question is actually relating to the identifiability of mixtures: if 
$$\rho_0 f(x;\theta_{00})+(1-\rho_0) f(x;\theta_{01}) = \rho_1 f(x;\theta_{10})+(1-\rho_1) f(x;\theta_{11})$$ 
this means that two mixtures produce the same sampling distribution, i.e., that the mixture is not identifiable. There exists a consequent literature on the topic, from Teichler (1961) who shows that a mixture is identifiable if and only if there exist $x_1$, and $x_2$ such that
$$\left|\begin{matrix} F(x_1;\theta_{00}) &F(x_1;\theta_{01})\\F(x_2;\theta_{00}) &F(x_2;\theta_{01})\\\end{matrix}\right|\ne 0$$
As a consequence, he shows that all Gaussian and Gamma mixtures are identifiable. Therefore, in this settings, imposing the condition
\begin{equation}
F_{01}(c)\times \rho_0+F_{00}(c)\times (1-\rho_0)=F_{11}(c)\times \rho_1+F_{10}(c)\times (1-\rho_1)
\end{equation}
implies that the components are equal two by two, e.g.
$$\rho_0=\rho_1\quad F_{01}=F_{11}\quad F_{00}=F_{10}$$
or
$$\rho_0=1-\rho_1\quad F_{01}=F_{10}\quad F_{00}=F_{11}$$
This is thus no an issue with MCMC but with the constraints themselves.
