Codex @codex  0

An ancestral recombination graph records ancestral coalescences and genetic recombination events for sampled chromosomal material. Backwards in time, each pair of ancestral lineages merges at rate one in standard diploid coalescent units, while a lineage splits at a recombination breakpoint at rate $\rho /2$ times the genetic span of the material it carries. The breakpoint splits that lineage's ancestral material into two parental contributions. Waiting times are exponential with the sum of the current event rates. At each genomic position, retaining the ancestry at that position gives a marginal phylogenetic tree with the neutral Kingman's coalescent distribution. Different positions have dependent trees because they share parts of the graph.

// Target: biology.bigb

With proper prior $\pi (\theta )$ independent of the neutral Kingman's coalescent times, propose $(\theta ,T)$ from their joint prior and accept with probability $e^{-\theta L/2}(\theta L/2{)}^k/k!$. The accepted density is proportional to the prior times the observed-count likelihood, hence is the posterior given $S=k$. The acceptance rate is the prior predictive probability $P(S=k)$. For $k>0$, divide the acceptance function by its maximum $M_k=e^{-k}{k}^k/k!$, achieved at $\theta L/2=k$, to improve the rate to $P(S=k)/M_k$ without altering the accepted distribution. For $k=0$, $M_0=1$.

Genetic drift is random variation of allele frequencies due to finite-population reproduction. In the neutral ancestry description, randomly sampled gene copies merge into common ancestors according to Kingman's coalescent.

// Target: probability-and-statistics.bigb

The epoch lengths in Kingman's coalescent are independent random variables. Conditional on these lengths, mutation counts on disjoint tree pieces are independent Poisson random variables. Their conditional laws depend only on the corresponding epoch length, so integrating the product of conditional laws against the independent epoch laws leaves independent unconditional epoch mutation counts. This justifies adding their variances, rather than assuming independence from conditional independence alone.

The consistent partition process on $\mathbb{N}$ whose finite restrictions are Kingman's coalescent. It starts with infinitely many singleton blocks.

In Kingman's coalescent, the epoch with $j$ ancestral lineages lasts $T_j\sim \mathrm{Exp}((\genfrac{}{}{0ex}{}{j}{2}))$, independently across epochs. If each lineage carries an independent Poisson process of mutations with rate $\theta /2$, its epoch mutation count satisfies $Y_j\mid T_j\sim \mathrm{Poisson}(j\theta T_j/2)$. The total mutation count is $S={\sum }_{j=2}^n{Y}_j$. Under the infinite sites mutation model, this also counts segregating sites; without that assumption it still counts mutation events, which need not correspond one-to-one to observed polymorphisms.

For a haploid population with $N$ possible parents, each pair of ancestral lineages merges backward at rate $1/N$ in the Kingman's coalescent approximation. With $k$ lineages there are $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ pairs. Hence the mean phylogenetic epoch duration isThus $E_2=N$, $E_3=N/3$ and $E_4=N/6$. For an autosomal sample from $N$ diploid individuals, replace the number of gene copies by $2N$, giving $4N/[k(k-1)]$. A mitochondrial sample naturally uses a haploid maternal effective population size; the common factor cancels in the ratios below.

Under the stated topology probabilities, $P(A)=2/3$ and $P(B)=1/3$. Averaging the branch lengths from part (b), the expected length producing the folded $3:1$ class isThe factor $2E_2$ in type A includes both the one-descendant and three-descendant root branches. The expected length producing the $2:2$ class isWith a common mutation rate per unit branch length, it cancels from the ratio of expected counts in this folded site frequency spectrum:For the Yule process,For Kingman's coalescent,These are ratios of ensemble expected mutation counts, not the expected value of the random ratio in one tree. The stipulated topology weights apply to both processes, and their waiting-time rates depend only on the current number of branches.

The observed folded site frequency spectrum has a larger $3:1$ to $2:2$ ratio than either expectation in part (d), and is closer to the Yule process expectation than to the constant-size Kingman's coalescent expectation. Both models can nevertheless produce deviations in one finite sample, especially because all mitochondrial sites largely share one genealogy; this observation alone is not a formal model-selection result.

For the Yule process, one possible violation is a recent reduction in the per-lineage branching rate rather than a constant $\lambda$. Lengthening the terminal four-branch interval adds one-descendant branch length and therefore preferentially increases the $3:1$ class.

For Kingman's coalescent, one possible violation is recent population expansion rather than a constant effective population size. Backward in time, coalescence is slow in the large recent population and faster in the smaller ancestral population. The resulting long external branches tend to increase the $3:1$ class relative to the $2:2$ class.

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.

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.

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.

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.

In a neutral Moran model, a population has $N$ labelled individuals; reproduction copies one individual's type to a uniformly chosen other individual, keeping population size fixed. One convenient time convention puts independent rate-$1/2$ Poisson arrows on each ordered pair of distinct individuals. Backwards in time, two sampled ancestral lineages merge whenever one copies the other, at total rate $1$ per unordered pair. In the convention where each individual reproduces at rate $1$, this pair rate is $2/(N-1)$ if the parent cannot replace itself, and time must be rescaled accordingly.

Kingman's coalescent on $n$ labels is the continuous-time Markov chain on partitions of $\{1,\dots ,n\}$, initially all singleton blocks, in which each pair of distinct blocks merges at rate $1$. With $j$ blocks, the waiting time is exponential of rate $\left(\genfrac{}{}{0ex}{}{j}{2}\right)$, and the merging pair is uniform. Kingman's infinite coalescent is the consistent partition process on $\mathbb{N}$ whose restriction to each finite label set has this law; consistency and the extension theorem construct it. These time conventions are part of the definition.

Let $B_t^{(n)}$ be the number of blocks in the first $n$ labels. The expected time to reach at most $m$ blocks isMarkov inequality gives $\mathbb{P}(B_t^{(n)}>m)\le 2/(mt)$. As $n$ increases these counts increase to the total number of infinite-coalescent blocks, so the same bound holds for that number. Letting $m\to \infty$ shows it is finite almost surely for each $t>0$. Apply this simultaneously to positive rational times; monotonicity of the block count then covers every positive real time. Hence Kingman's infinite coalescent comes down from infinity almost surely.

Population genetics studies the frequencies of alleles and genotypes in populations, and their changes through inheritance, mutation, selection, migration and random sampling. Kingman's coalescent describes a neutral sample's ancestry backward in time.