This question is essentially same as this one. The question is: How do you calculate conditional probability of a node in Bayesian network when it has a continuous node as a parent? However, I cannot understand its answer given by @steffen. My problems with the answer are as follows:
(I use the exact same variables as used in the previous question.
harvest are both continuous variables).
What is the justification in mapping the current value of
harvest to a parameter describing the distribution of
cost? For example,
$p(Cost=c|Harvest=h)=N(\alpha * h + \beta,\sigma_2^2)(c)$
for some $\alpha$ and $\beta$ and fixed $\sigma^2$.
I carried out an experiment to check how this works. I modeled
harvest as $N(\mu=10,\sigma^2=1)$. I modeled
cost as $N(\mu=2,\sigma^2=0.5)$. Now, I want $P(cost=10 | harvest=10)$ which should be close to $0$. Now if we take some value of $\alpha$ and $\beta$, say $1$ and $0$ (and $\sigma^2=1$), we get a high probability according to the above equation.
So it is clear that $\alpha, \beta$ and $\sigma^2$ play a major role. Now do I do some sort of fitting to determine these values? If yes, exactly how? Let's say I have 1000 values for
harvest, and I want to calculate $\alpha, \beta, \sigma^2$ such that it closely approximates the true $P(cost | harvest)$. (Let's assume that I did a very fine sampling of the space of cost and harvest to get true $P(cost | harvest)$ as a discrete conditional probability distribution).
I hope my understanding is right. If not, please correct it.