Consider an information source which produces the signals 1 and 0 with unknown probabilities. The output of the source is denoted by the discrete variable Y. Y=1 implies that the temperature of a process has exceeded a predefined threshold, whereas Y=0 implies that the process is under control (i.e., the temperature is below the threshold). The temperature threshold is represented by $\theta$. Assuming the process temperature (represented by X) follows a continuous distribution, how could $f_{X \mid Y=y}(x)$ be estimated?
My proposed procedure:
Using Bayes' rule:
$\begin{equation*}f_{X \mid Y=y}(x)=\frac{P[Y=y|X=x] f_{X}(x)}{P[Y=y]}\end{equation*}$
Where:
$P[Y=y] = \int_{x=-\infty}^{\infty} {P[Y=y|X=x]}{f_X(x)}dx$
Moreover, for the likelihood, possible representations (that I am considering) might be:
$P[Y=1|X=x]=\begin{cases}(1-p)/2, & x\leq a\cr p, & a<x< b\cr (1-p)/2, & x\geq b\end{cases}$
or:
$P[Y=1|X=x]=\begin{cases} p, & x=\theta \cr 1-p, & otherwise\end{cases}$
where a, b, and c are positive constants and p represents a probability estimate. Furthermore, $a<\theta< b$
Specific questions: I am new to probability theory, so I would appreciate any help with the following questions:
- Do any of the proposed likelihood representations make sense from a probabilistic perspective?
- If they are incorrect, what could be examples of alternative representations for $P[Y=y|X=x]$ in the context of the described situation?