# Finding the probabilities from density distribution to use in Bayesian formula

I have some search results, which have been validated based on a certain criterion and each hit has a probability of being correct assigned to it.

The search results look like this:

Var1---Probability---T

A-------0.1-----------28

B-------0.05----------37

C-------0.6-----------110.45

D-------0.0001------12.89

Each of these Var1s have another characteristic assigned to them (T) (unrelevant to the search engine's criterion) from which I can further validate the results. I have a library which contains the expected Ts for each of Var1s and from there I can tell if a hit is correct based on how close it's T is to the expected T for that hit.

So, to update the search results probabilities, according to the new criterion, I am going to use a Bayesian probability like this:

P(Var1|T)= (P(T|Var 1)*P(Var1)) / (P(T))

Here is my problem:

For finding P(T|Var 1) and P(T), I should figure out the specific probabilities for T=x. I have to figure these out from the density distributions. Basically, the area under the density distribution plot in a certain interval is the probability of that region but for an individual point the probability will be zero by definition because there would be no area for integration.

So to find P(T=x|Var 1) and P(T=x) I have to consider an interval. Selecting that interval is tricky because the smaller the interval, the smaller would be the probabilities!

So, my question is that what should I do in this case, when I want to find the probability of a specific observation from the density distribution? Is there any other way to find the probability of a point when I have the distribution? If I have to consider an interval for finding the probability, what should be the length of that interval?

The distributions that I have for P(T|Var 1) and P(T) are as follows:

P(T|Var 1): for any of the Var1’s I have a distribution of the expected (and almost confidently correct) T’s in my library.

P(T): I have the distribution of all T’s.

P(Var1): This is the initial probability assigned to each of Var1s based on the first criterion.

Thank you very much and please let me know if you needed more clarification.

If your variable $T$ is continuous, you can just plug the value of the marginal and conditional densities, $p(T)$ and $p(T|\mathrm{Var1})$, directly into Bayes' formula (assuming you have a way to evaluate these).