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Given are multiple probability distributions for a random variable.

Is there a way to calculate the most probable value based on these distributions?

Example:

Random Variable: Number of hairs on the head of a person Probability Distribution 1: Number of hairs depending on gender Probability Distribution 2: Number of hairs dependent on age Probability Distribution 3: Number of hairs dependent on hair color  Now I want to calculate the most probalble value for the number of hairs for a 34 years old, blonde woman. Does this sound reasonable? 
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    Ok, I reformulated the question. The example should demonstrate the problem well in my eyes.2012-05-08

1 Answers 1

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It sounds like you are in the realm of conditional probability. In particular, let $N$ be the number of hairs, $G$ the gender, $A$ the age and $H$ the hair color. You have the following distributions:

$P(N|G)$, $P(N|A)$ and $P(N|H)$.

You are interested in $P(N|G, A, H)$. The issue is that you don't quite have enough data to make observations. What you really need is how gender age and hair color correlate with each other. For example, you can assume that age and gender are independent. As well, maybe gender and hair color are independent as well. Assuming you don't count gray hair, age and hair color can be considered independent too.

From the probability chain rule:

$P(N|G,A,H) = \frac{P(N,G,A,H)}{P(G|A,H)P(A|H)P(H)}$

Now the terms in the denominator by independence can reduce to $P(G)$, $P(A)$ and $P(H)$ each of which you can determine just by looking at statistics, say $P(G)$ is just $1/2$ for male female, etc. For the numerator you just need to take a large sample of people and write down the number of occurances of each quadripule $(N,G,A,H)$.