I have two probability density functions such that
$$1 = \int_{a}^{b}\text{pdf}_{1}(x)dx = \int_{a}^{b}\text{pdf}_{2}(x)dx $$
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 < c $ where $a < c < b$ and more confident in values of $\text{pdf}_{2}$ 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.
This sounds like a job for mixture distributions. Define a measurable function $\phi: [a,b] \rightarrow [0,1]$ to represent your "confidence" that the first distribution is applicable (i.e., the probability that this distribution is applicable). Now form the mixture distribution:
$$p_* (x) \equiv \phi(x) \cdot \text{pdf}_1(x) + (1 - \phi(x)) \cdot \text{pdf}_2(x).$$
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 $\phi$.