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Natural selection involves systematic differences in expected reproductive success associated with variation in phenotype. For a continuing evolutionary response, the relevant variation must be inherited: offspring must tend to preserve the differences being favored. Actual offspring counts also fluctuate by chance, producing genetic drift. Survival matters insofar as it affects contribution to future generations. Genotype fitness is therefore contextual, depending on environment and the life cycle being counted, rather than simply measuring strength or longevity. See Clark, Douglas and Choi, Biology 2e: population evolution.

In chapter IV of his 1859 book, Darwin explained natural selection through preservation of favorable variations and greater chances of survival and reproduction. See Darwin, On the Origin of Species, first edition, chapter IV. The beautiful consequence is cumulative change without foresight: inherited differences are filtered repeatedly by present conditions. Natural selection neither generates a needed mutation on demand nor guarantees progress toward a predetermined goal.

During the epoch with $j$ ancestral lineages, Kingman's coalescent has waiting time $T_j\sim \mathrm{Exp}((\genfrac{}{}{0ex}{}{j}{2}))$. In the stated time units, each lineage accumulates mutations at rate $Nu=\theta /2$, so the mutation count on a neutral coalescent tree decomposes asThe law of total expectation and the mean of an exponential distribution give $\mathbb{E}{Y}_j=(j\theta /2)/\left(\genfrac{}{}{0ex}{}{j}{2}\right)=\theta /(j-1)$. ThereforeThis calculation counts mutation events. Equating the count with observed segregating sites additionally uses the infinite sites mutation model.

Linkage disequilibrium means that alleles at different genetic loci are associated in population haplotypes. For two biallelic loci,The first quantity measures departure from independent allele frequencies; the second is the squared allelic correlation. Standardized linkage disequilibrium $D^{\prime }$ instead compares $D$ to its possible range at the given frequencies. High $|D^{\prime }|$ can coexist with low $r^2$, so it need not make one genetic marker a good predictor of another. Genetic linkage is a different notion: it describes within-family transmission and the recombination fraction.

The chromosome-21 study of Patil and colleagues, Science 2001 found consecutive single-nucleotide polymorphisms organized into regions with comparatively few common haplotypes; typically three haplotypes accounted for over 80% of the sampled chromosomes within a block. Such haplotype blocks can be represented efficiently by tag SNPs. Strong association within blocks and weaker association across boundaries give a useful description, though block boundaries depend on allele frequencies, population history, sampling and the operational definition of a block. They do not divide chromosomes into regions where genetic recombination is respectively impossible and certain.

The ancestry explanation uses an ancestral recombination graph. Going backwards, ancestral lineages coalesce, while a genetic recombination event splits one lineage's ancestral material into contributions from two parents. At a fixed genomic position, retaining only its ancestry gives a phylogenetic tree with the appropriate coalescent law. Nearby positions often retain much of the same tree; mutations on shared branches then produce correlated allele states. Breakpoints can change the local tree, weakening that shared history.

Under a simple random-mating two-locus model, linkage-disequilibrium decay under recombination gives $D_{t+1}=(1-r)D_t$. Actual populations also undergo genetic drift, population bottlenecks, mutation and migration, which can create or preserve association. Recombination rates themselves vary along chromosomes: McVean and colleagues, Science 2004 inferred strong local variation from sequence data. These observations motivate an ancestry model with spatially varying recombination, rather than treating physical distance as the only determinant of linkage disequilibrium.

For fine mapping, a typed genetic marker can show genetic association because it is correlated with a nearby causal genetic variant. Linkage-disequilibrium marker tagging makes a sparse first-stage scan possible, but several variants in one block can produce nearly indistinguishable association patterns. Historical recombination and diverse local genealogies can narrow the interval when denser genotyping is available.

An ancestral recombination graph turns that intuition into a likelihood calculation. One can place a proposed disease-origin event on a local ancestral branch and integrate over possible graphs, mutation histories and unobserved haplotypes. The local phylogenetic tree then relates disease-bearing chromosomes and predicts which markers should remain associated. In a Bayesian inference, uncertainty in ancestry, mutation location and demographic parameters enters the posterior distribution rather than being replaced by one guessed tree. Whole-graph inference is computationally demanding, and approximate or pairwise methods often supply practical estimates.

The useful distinction is between finding an associated region and identifying its causal variant. Strong linkage disequilibrium improves marker coverage but can limit resolution inside the associated region. Population stratification, ascertainment, rare variants and misspecified population history can also mislead an association analysis. Combining fine mapping with independent biological evidence is therefore stronger than interpreting an association peak as proof of causation.

Assume the parents are unaffected, the disease has complete penetrance with a single recessive causal allele, and there are no phenocopies, new disease mutations or alternative causal loci. An affected first child has genotype $aa$, so each unaffected parent must be a heterozygote $Aa$. Conditional on these parental genotypes, Mendelian segregation in the second child's paternal and maternal meioses is independent of the transmissions to the first child. ThereforeThis is the same sibling recurrence risk as for unrelated, unaffected parents under the same recessive model. Parental kinship increases the prior probability that both parents carry the same rare disease allele; after both are identified as carriers, it does not alter their Mendelian transmission probabilities. The factor $63.4$ from the preceding part is therefore not an additional multiplier on the sibling recurrence risk.

If unaffected parents are not explicitly required, the rare-disease assumption is an approximation supporting the usual carrier-by-carrier calculation, rather than a logically exact consequence of an affected child alone. Under complete penetrance, the compatible parental matings are $Aa\times Aa$, $Aa\times aa$ and $aa\times aa$, with second-child risks $1/4,1/2,1$ respectively. Let their prior probabilities be $w_{11},w_{12},w_{22}$, including both orders in $w_{12}$. Conditioning on the affected first child givesWithout negligible or excluded affected-parent matings, this mixture need not equal $1/4$ and may depend on consanguinity. Likewise incomplete penetrance or disease heterogeneity invalidates the assertion that an affected child of unaffected parents necessarily identifies an $Aa\times Aa$ mating. The stated $1/4$ answer uses the unaffected-parent, fully penetrant single-locus assumptions.

Let $t=(t_2,\dots ,t_n)$, with all $t_j>0$, and put $L(t)={\sum }_{j=2}^{n}j{t}_j$. The independent exponential epoch times have joint densityTake a proper prior distribution density $\pi (\theta )$ on $\theta \ge 0$, independent of the neutral genealogy. Independence is appropriate because the specified neutral Kingman's coalescent does not depend on the mutation parameter. The conditional observed-count likelihood function isIts prior predictive probability, or Bayesian model evidence, isAssuming $m_k>0$, Bayes' theorem gives the joint posterior of mutation rate and coalescent times:It is zero outside the prior support and positive-time region. For $k=0$, use the usual convention $0^0=1$ in the Poisson mass. An improper prior cannot be sampled as the proposal in the following algorithm, even in cases where its formal posterior can be normalized.

Let $\mu$ be the mutation rate per chromosomal segment per generation, and use the population-scaled parameter $\theta =4N_e\mu$. In $2N_e$-generation coalescent units, mutations occur on each ancestral lineage at rate $\theta /2$. Independent mutations on all branches of a given phylogenetic tree superpose to give a Poisson distribution with mean $\theta L/2$.

Under the infinite sites mutation model, each mutation occurs at a distinct position, and every mutation below the sample's common ancestor separates a nonempty proper subset of sampled chromosomes from the remainder. Thus the mutation count is exactly the segregating site count, givingBy the law of total expectation and the previous branch-length formula,This also gives the Watterson estimator $S/a_n$ as an unbiased moment estimator of $\theta$. Here all sample segregating sites are counted, including singletons; filtering sites by a minimum sample frequency or by external SNP discovery would introduce genetic-study ascertainment and change this calculation.

Use standard diploid coalescent time units of $2N_e$ generations, where $N_e$ is the fixed effective population size, and take $n\ge 2$. In the neutral Kingman's coalescent, each unordered pair of ancestral lineages merges at rate one. With $j$ lineages, the next coalescence time therefore has an exponential distribution with rateThe memoryless construction and the Markov property make these epoch durations independent. During epoch $j$ there are $j$ branches, each of duration $T_j$, so the total branch length of a neutral coalescent isBy linearity of expectation,The ancestral branch above the sample's most recent common ancestor is excluded: mutations there would be shared by all sampled chromosomes and would not create segregating sites.

Suppose a disease-risk mutation first occurs on an ancestral haplotype. Descendants carrying the risk allele initially share that surrounding haplotype. Historical genetic recombination progressively replaces the flanks while retaining the disease allele, leaving linkage disequilibrium between the causal variant and some nearby genetic markers. A noncausal marker can therefore have different allele frequencies in affected individuals and suitably chosen controls. This is the basis of association mapping.

One first tests dense markers or multi-marker haplotypes for genetic association with the disease phenotype. Strongly correlated single-nucleotide polymorphisms can be represented by tag SNPs to locate an associated region efficiently. Within that region, denser genotyping and examination of disease-associated haplotypes locate historical recombination breakpoints: segments retained across independently recombined disease-bearing chromosomes are stronger candidates for containing the causal allele. Compare affected and unaffected carriers as well, since incomplete penetrance and other risk factors can prevent a perfect haplotype-disease correspondence.

Association mapping can achieve fine resolution because it uses recombination accumulated over many generations, rather than only the few observed meioses in a present-day linkage pedigree. Resolution depends on the local recombination history: a long region of strong LD may identify a disease-associated block while leaving many variants statistically indistinguishable. Additional informative haplotypes or populations with different LD can help separate them. Conversely, very weak LD requires closer marker spacing to avoid missing the causal region.

Population stratification must be controlled by ancestry-matched comparisons, statistical adjustment or an appropriate family-based association design, since ancestry can affect both disease frequency and marker frequency. A marker association alone establishes neither causality nor identity of the disease gene; nearby correlated variants must be distinguished by further genetic and functional evidence. Use LD to detect a disease-associated ancestral segment, then use historical recombinants to narrow the causal interval. The correlation between causal variants and their surrounding haplotypes, together with the value of representative markers, is illustrated in the HapMap study.

Across a human chromosome, neighboring single-nucleotide polymorphisms often show substantial linkage disequilibrium, while average allelic correlation tends to decline with increasing genetic distance. A new mutation starts on one ancestral haplotype; repeated genetic recombination separates it from progressively more distant markers. In an ideal randomly mating two-locus population without further mutation, migration or selection, gamete formation gives $p_{11}^{\prime }=(1-c)p_{11}+c{p}_A{p}_B$, where $c$ is the recombination fraction. The marginal allele frequencies remain fixed, soHere the prime means the next-generation value, not a separately standardized LD statistic. This simple decay explains the distance trend, rather than asserting an exact deterministic law for real finite populations.

The local pattern is heterogeneous. Many regions have haplotype blocks with strong LD and a small number of common haplotypes, separated by intervals showing more historical recombination, often recombination hotspots. A hotspot can therefore cause a rapid drop in LD between markers that are close in physical distance. Block lengths and boundaries are not uniform, perfectly sharp or independent of the population and the statistical block definition. These empirical features are documented in the HapMap data.

Population history also matters. Population bottlenecks can reduce the number of ancestral haplotypes and increase the extent of LD; population stratification or admixture can create longer-range associations, including between distant loci. Different populations can consequently have different haplotype frequencies and LD patterns at the same genomic interval. Raw $D$ also varies with allele frequency, so comparing LD across markers requires attention to the chosen measure. The typical pattern is locally strong, block-like and highly variable LD, with a broad decline over genetic distance rather than a uniform chromosome-wide decay.

An ancestral recombination graph, or ARG, is the directed ancestry of a sample of chromosomal segments when ancestral lineages can both coalesce and undergo genetic recombination. It records which ancestral material each lineage carries, the times of events and the recombination breakpoints. At any fixed genomic position, following only the ancestral material at that position gives a marginal phylogenetic tree. Different positions can have different trees, while sharing substantial parts of their history. The graph is acyclic in time, although its underlying undirected graph can have loops because a lineage splits backwards and its ancestors can later rejoin.

For the standard neutral, constant-size, randomly mating diploid model, measure time in $2N_e$ generations and set $\rho =4N_{e}r$, where $r$ is the recombination rate across the whole segment. The backward process is a continuous-time Markov jump process on collections of lineages carrying ancestral material:

For a state with $k$ lineages, independent exponential event clocks give total rateThe next waiting time has exponential distribution with rate $R$; choose its event type in proportion to the corresponding rate. A coalescing pair is uniform among unordered pairs; a recombining lineage is selected with weight $b_i$ and its breakpoint is uniform over its active span. With a nonuniform genetic map, use the corresponding integrated intensity and map-weighted breakpoint distribution. Initially all sample lineages span the whole segment, so the initial rate is $\left(\genfrac{}{}{0ex}{}{n}{2}\right)+n\rho /2$. Ancestral material no longer needed after its sample copies have found their local common ancestor can be pruned.

Mutations are placed independently on the resulting ancestral branches, with local rate $\theta /2$ in coalescent units for a locus having scaled mutation parameter $\theta$. Without recombination, the graph reduces to a single Kingman's coalescent tree. With recombination, each individual locus still has that marginal coalescent distribution, but trees at different loci are dependent. The ARG combines backward pairwise mergers with backward recombination splits; local trees are its position-specific projections. The neutral two-locus construction is developed in Hudson's two-locus model.

Five consequences of the ancestral recombination graph are:

Shared ancestry explains correlation; recombination makes ancestry a mosaic; mutation on the resulting branches converts genealogy into observed variation. These explanations use the neutral ARG, with a variable recombination map where stated. Selection, population subdivision or changing population size can also be studied genealogically, but require extending the event model rather than being consequences of the constant-size panmictic neutral process alone.