I have access to M probability measures, $P_e(c_1),P_e(c_2),\cdots,P_e(c_M)$, defined as \begin{equation} P_e(x) = p(x|y) = p(y|x)\cdot \mathbb{P}(X=x) \frac{1}{\sqrt{2\pi\sigma^2}} \exp\Big[-\frac{(y-\alpha x)^2}{2\sigma^2}\Big] \mathbb{P}(X=x), \end{equation} where $x\in\{c_1,c_2,\cdots,c_M\}$ and is discrete. $p(x|y)$ is a PMF
I need to estimate $\sigma^2,\alpha,y$ from these M probabilities.I understand that this is an overdetermined system and therefore has no consistent solution. But how can I find a reasonable estimate of these quantities.
Side note: Additional constraints that might be helpful are:
- $\sum _{i=1}^{M} P_e(c_i)=1 $
- $\alpha>0$
- $\sigma\in\mathbb{R}, \sigma>0$
Since I am very bad with probabilities, this answer could be totally stupid and, depending on comments, I should delete it.
It seems to me that, having $n$ data points $(x_i,P_i)$, the problem is relevant from nonlinear regression; one of the issues is to first obtain reasonable estimates for the parameters $\sigma^2,\alpha,y$. So, take logarithms first and rearrange the expression. We then have $$\log(P)=-\left(\log(\pi \sigma^2)+\frac{y^2}{\sigma^2}\right)+\frac{2\alpha y}{\sigma^2} x-\frac{\alpha^2}{\sigma^2}x^2$$ Defining $z=\log(P)$, we then face a quadratic model $$z=A+B x+Cx^2$$ which is easy to fit using ordinary least square method $(z_i=\log(P_i))$. When parameters $(A,B,C)$ have been found we then have to solve $$C=-\frac{\alpha^2}{\sigma^2}\tag1$$ $$B=\frac{2\alpha y}{\sigma^2}\tag2$$ $$A=-\left(\log(\pi \sigma^2)+\frac{y^2}{\sigma^2}\right)\tag3$$ which can easily be solved using successive eliminations.
Using these estimates, you have all elements for starting a nonlinear regression (you need to do it since what is measured is $P$ and not any of its possible transforms such as $\log(P)$.
If , for any reason, you do not want (or cannot) use nonlinear regression, you could use this method extending it to three parameters (this is simple to do).
Now, the problem is the constraint. If you need to take it into account, I am afraid that you would need a nonlinear optimization (the objective function being the sum of the squares of the residuals) able to handle constraints.