I have two probability density functions such that 1 = ∫ a b pdf 1 ( x ) d x = ∫ a b pdf 2 ( x ) d xwhich 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 &lt; c where a &lt; c &lt; b and more confident in values of pdf 2 where c &lt; 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.

Question

I have two probability density functions such that  1  =      ∫          a              b            pdf          1        (  x  )  d  x  =      ∫          a              b            pdf          2        (  x  )  d  xwhich 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  &amp;lt;  c where   a  &amp;lt;  c  &amp;lt;  b and more confident in values of       pdf          2       where   c  &amp;lt;  x. Is it possible to combine these pdfs to one while keeping this &quot;confidence&quot; scheme in mind? (Assuming that I have only access to the pdfs - not an assumption on the underlying process).I can definitely &quot;average/combine&quot; 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&#039;m unsure if I can really proceed post-measurement.

wasipewelr · Accepted Answer

This sounds like a job for mixture distributions. Define a measurable function   ϕ  :  [  a  ,  b  ]  →  [  0  ,  1  ] to represent your &quot;confidence&quot; 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.