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, classical approach of multiplying the densities that was called conflation by Hill & Miller in 2011. ^{[1] [2]}

Conflation

Conflation refers to the merging of independent probability density functions using simple multiplication of the constituent densities. ^[1] Similar methods, with narrower definition ar known in the literature as aggregating, amalgamating, blending, coalescing, combining, composing, compounding, conjoining, conjugating, consolidating, converging, convoluting, incorporating, integrating, joining, lumping, mixing, pooling, unifying, uniting. ^[2]

Coalescence

Coalescence is opposed to the "Multiplication Rule" (also called "Merging by Multiplication"), which disregards that the probability of occurrence in each frequency class changes proportionally to the probability reference base accumulated in the considered class. ^[2] Simple 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 & Miller 2011.
2. Van Droogenbroeck, Frans J., 'Coalescence, unlocking insights in the intricacies of merging independent probability density functions' (2025).

Sources

• Hill, Theodore P.; Miller, Jack (2011-09-01). "How to combine independent data sets for the same quantity". Chaos: An Interdisciplinary Journal of Nonlinear Science. 21 (3) 033102. arXiv: 1005.4978 . Bibcode:2011Chaos..21c3102H. doi: 10.1063/1.3593373 . ISSN   1054-1500. PMID   21974637.