I have two probability density functions such that 1 = ∫ a b pdf 1...

Carly Cannon

Carly Cannon

Answered

2022-07-06

I have two probability density functions such that
1 = a b pdf 1 ( x ) d x = a b pdf 2 ( x ) d x
which represent the same underlying process (i.e. obtained from different measurements).
I am more confident in the values of pdf 1 for values of x < c where a < c < b and more confident in values of pdf 2 where c < x. Is it possible to combine these pdfs to one while keeping this "confidence" scheme in mind? (Assuming that I have only access to the pdfs - not an assumption on the underlying process).
I can definitely "average/combine" the pdfs as a whole, but is it possible to be more confident in a section? The value of a range of the pdf is really meaningless without knowing the rest of the distribution, so I'm unsure if I can really proceed post-measurement.

Answer & Explanation

wasipewelr

wasipewelr

Expert

2022-07-07Added 11 answers

This sounds like a job for mixture distributions. Define a measurable function ϕ : [ a , b ] [ 0 , 1 ] to represent your "confidence" that the first distribution is applicable (i.e., the probability that this distribution is applicable). Now form the mixture distribution:
p ( x ) ϕ ( x ) pdf 1 ( x ) + ( 1 ϕ ( x ) ) pdf 2 ( x ) .
This gives you a new density that combines the two density functions you were initially working with. The new density function is a weighted average of the original densities based on your function ϕ.

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