Coalescence (statistics)

For other uses, see Coalescence (disambiguation).

In statistics, coalescence refers to the merging of independent probability density functions. The systematically recommended Multiplication Rule (also known as conflation) for merging densities generates a joint density that suffers from a mean-biased expected value and an overly optimistic standard deviation. The Multiplication Rule disregards that the probability of occurrence in each frequency class changes proportionally to the probability reference base accumulated in the considered class. The overlooked intricacy requires a substantial probabilistic adjustment that the simple Multiplication Rule cannot possibly provide. The conditional nature of the issue imposes an elementary Kolmogorovian-Bayesian reassessment.

Theorem

The coalesced density function d(x) of n independent probability density functions d₁(x), d₂(x), …, d_k(x), is equal to the reciprocal of the sum of the reciprocal densities:

${\begin{aligned}{\frac {1}{d(x)}}&={\frac {1}{d_{1}(x)}}+{\frac {1}{d_{2}(x)}}+\cdots +{\frac {1}{d_{n}(x)}}\\[5mu]\end{aligned}}$