#Theta and the sub-population correction

#The problem it fixes

The product rule - multiply 2 pa pb across loci - assumes the reference population is one big randomly-mating pool. Real populations are not. They are made of sub-populations that mate within themselves more often than across, and two people drawn from the same one share alleles more often than the overall frequencies predict.

Ignore that and the match probability comes out too small: the profile looks rarer than it is, and the evidence looks stronger than it is. Theta is the correction.

#What theta is

Theta (θ, also written Fst) is the co-ancestry coefficient: the probability that two alleles drawn from the same sub-population are identical by descent rather than by coincidence. It is a single number standing in for the whole population structure.

Larger theta means more shared ancestry, which means a larger - more conservative - match probability.

#The formulae

NRC II recommendation 4.2, the Balding-Nichols sampling formula, conditioned on one copy of the genotype already being observed. That conditioning is what makes it a match probability rather than a population frequency:

homozygote  aa:  [2θ + (1-θ)pa] · [3θ + (1-θ)pa] / [(1+θ)(1+2θ)]

heterozygote ab: 2 · [θ + (1-θ)pa] · [θ + (1-θ)pb] / [(1+θ)(1+2θ)]

At θ = 0 both collapse to the familiar Hardy-Weinberg forms, pa² and 2 pa pb. The deck implements exactly this:

forensics.genotypeFrequency($db, $g, $theta);

Theta must be in [0, 1). The interval is half-open because at θ = 1 every allele is identical by descent and the formula stops meaning anything, even though it still divides.

#Which value

Three named constants, because these are the choices that actually get made:

Any float in [0, 1) works; the constants just name the conventional ones.

#What it does to a number

From examples/matchprobability.j, an eight-locus profile against the bundled synthetic table:

Three orders of magnitude between no correction and the isolated-population value, on the same profile and the same table. Theta is not a rounding detail - it is a stated assumption about the population, and it belongs in anything you write down alongside the number it produced.

#Where theta is applied - and where it is not

Applied: the single-source match probability. genotypeFrequency, randomMatchProbability, log10RandomMatchProbability and matchLikelihoodRatio all take a theta and use it.

Not applied - by design:

• inclusionProbability, cpi, cpe. The probability of inclusion counts who cannot be excluded; it is not a co-ancestry-corrected match probability, and applying theta to it would be mixing two different questions. See Mixtures.
• kinship.j and pedigree.j. These use the plain product rule throughout. The sub-population correction does not extend to the IBD-1 term - the joint probability of two related people's genotypes under co-ancestry needs the full recursive sampling formula, not a swapped-in allele frequency. Rather than apply theta to some terms and not others and produce a number that is neither one thing nor the other, those modules are uniformly theta-free and say so.

This is a real limitation, not a preference; it is listed as one in Limitations. It also matches what other relationship-testing software does by default.

#Theta and the frequency floor are different things

Both make an estimate more conservative, and they are easy to confuse:

• the 5/2N floor handles sampling - the reference table is finite, so a rare allele's frequency is not well measured. See Reference frequency data.
• theta handles structure - the population is not one pool, so alleles correlate within it.

They compose: alleleFrequency applies the floor, and genotypeFrequency applies theta to whatever the floor returned.