Coalescence (statistics)

For other uses, see Coalescence (disambiguation).

In statistics, coalescence refers to the merging of independent probability density functions. It contrasts with the simpler, erroneous approach called conflation.

Conflation

Conflation refers to the merging of independent probability density functions using simple multiplication of the constituent densities.^[1] 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.

Coalescence

Unfortunately, conflation generates a joint density that suffers from a mean-biased expected value and an overly optimistic standard deviation. The conditional nature of the issue imposes an elementary Kolmogorovian-Bayesian reassessment.^[2] This shortcoming is satisfactorily solved by the coalescense method.

Coalesced density function

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:

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

References

1. Hill Th. P., Miller J., Fox R. F., ‘How to Combine Independent Data Sets for the Same Quantity’, Chaos (Woodbury, 2011) 1-20.
2. Van Droogenbroeck, Frans J., 'Coalescence, unlocking insights in the intricacies of merging independent probability density functions' (2025).