coalescence (statistics)
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{{other uses|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.Hill Th. P., Miller J., Fox R. F., [https://www.researchgate.net/publication/51698644/ ‘How to Combine Independent Data Sets for the Same Quantity’, Chaos (Woodbury, 2011) 1-20.] 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.Van Droogenbroeck, Frans J., [https://www.academia.edu/127477986/ 'Coalescence, unlocking insights in the intricacies of merging independent probability density functions'] (2025). 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), …, dk(x), is equal to the reciprocal of the sum of the reciprocal densities:
:
\frac{1}{d(x)} &= \frac{1}{d_1(x)} + \frac{1}{d_2(x)} + \cdots + \frac{1}{d_n(x)} \\[5mu]
\end{align}