Correct Application of Conditional Probabilities

Suppose you have a multivariate probability distribution function for 4 variables X1, X2, X3, X4 : P(X1, X2, X3, X4)

Normally, you can write P(X1|X2 = x2, X3 = x3, X4 = x4) : Then, you can find also find out the "most probable value" (i.e. expectation) of X1|X2 = x2, X3 = x3, X4 = x4. This can either be done using analytical integration for tractable integrals, or numerical sampling methods for intractable integrals (e.g. Monte Carlo Sampling).

My Question: Using the laws of probability, can one of these variables be taken out all together?

For example, suppose you are only given P(X1, X2, X3, X4) - is it possible to find out P(X1|X2 = x2, X4 = x4)? Can you find out the "most probable value" of X1 (i.e. expectation) : X1|X2 = x2, X4 = x4?

I.e. If you only have P(X1, X2, X3, X4), can you somehow determine P(X1, X2, X4)?

Or is this impossible?

Thanks!

Solution 1:

I am not really sure what you are asking, but if you have the joint distribution of all the data, you can recover any other distribution of interest (marginals, conditionals, etc.).

For instance, in the discrete case, by the law of total probability,

$\sum_{x_3}P(x_1,x_2,x_3,x_4)=P(x_1,x_2,x_4)$.