Carly Cannon

2022-07-06

I have two probability density functions such that
$1={\int }_{a}^{b}{\text{pdf}}_{1}\left(x\right)dx={\int }_{a}^{b}{\text{pdf}}_{2}\left(x\right)dx$
which represent the same underlying process (i.e. obtained from different measurements).
I am more confident in the values of ${\text{pdf}}_{1}$ for values of $x where $a and more confident in values of ${\text{pdf}}_{2}$ where $c. 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.

wasipewelr

Expert

This sounds like a job for mixture distributions. Define a measurable function $\varphi :\left[a,b\right]\to \left[0,1\right]$ to represent your "confidence" that the first distribution is applicable (i.e., the probability that this distribution is applicable). Now form the mixture distribution:
${p}_{\ast }\left(x\right)\equiv \varphi \left(x\right)\cdot {\text{pdf}}_{1}\left(x\right)+\left(1-\varphi \left(x\right)\right)\cdot {\text{pdf}}_{2}\left(x\right).$
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 $\varphi$.

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