Local Ancestry Along Admixed Chromosomes

Why Local Ancestry Is a Chromosome-Level Question

An admixed person's ancestry need not be the same along every part of a chromosome. Recombination over generations can leave one segment with ancestry from one source population and a neighboring segment with ancestry from another. At any particular genomic location, however, ancestry is not observed directly: we observe genetic markers whose alleles are more or less compatible with the different source populations.

A hidden Markov model (HMM) formalizes this problem. It treats ancestry at successive markers as a sequence of unobserved states and the observed alleles as evidence about those states. The model combines two kinds of information: how likely ancestry is to persist over the distance between markers, and how likely each observed allele is under each ancestry state.

The Markov assumption says that, given the ancestry state at the preceding marker, the next state does not additionally depend on earlier ancestry states. The emission assumption says that, given the ancestry state at a marker, its observed allele does not additionally depend on the other marker observations. These are modeling assumptions, not properties guaranteed by the data. In particular, simple allele-frequency emissions can be inadequate when marker dependence from linkage disequilibrium is substantial.

For a biallelic marker, write \(f_{kj}\) for the frequency of allele 1 at marker \(j\) in source population \(k\). In a simple model, the probability of observing allele 1 in state \(k\) is \(f_{kj}\), while that of observing allele 0 is \(1-f_{kj}\). This treats the source frequencies as fixed inputs and assumes the allele observation is appropriate for a single phased chromosome. Genotype likelihoods or unphased diploid data require emissions suited to those observations; the haploid formulas below should not be applied to them without modification.

Transitions From Recombination

A transition matrix describes how ancestry can change between adjacent markers. Its rows correspond to ancestry at the first marker and its columns to ancestry at the next; every row must sum to 1. In a simple pulse-admixture model, let \(q_k\) be the proportion of ancestry from source \(k\), let \(g\) be the number of generations since the admixture pulse, and let \(d_j\) be the genetic distance between markers \(j-1\) and \(j\), measured in Morgans. Under a Poisson crossover approximation, the probability of no ancestry-changing crossover over that interval is \(a_j=\exp(-g d_j)\). If a crossover occurs, the ancestry at the new position is drawn from proportions \(q_1,\ldots,q_K\). This gives the transition model

Here \(\mathbf{1}(k=l)\) equals 1 when the two labels match and 0 otherwise. A short interval has \(a_j\) near 1 and usually preserves ancestry; a longer interval or more generations since admixture allows more opportunity for ancestry to change. This is a simplified transition model. Its assumptions include a pulse of admixture, the specified ancestry proportions, and the crossover approximation; other demographic histories or transition models can be used when appropriate.

From Marker Evidence to Posterior Ancestry

For observed alleles \(x_1,\ldots,x_m\), the local ancestry probability at marker \(j\) is a posterior probability such as \(P(Z_j=k\mid X_1=x_1,\ldots,X_m=x_m)\). It combines the ancestry prior and transitions with the emissions at all observed markers. In particular, information from markers on both sides can inform the state at \(j\). This is often called a smoothed posterior, in contrast to a probability calculated using only observations up to marker \(j\).

A useful calculation is the forward recursion. Let \(\alpha_j(k)\) be the joint probability of the observations through marker \(j\) and state \(Z_j=k\). The initial value is the initial state probability multiplied by the first emission. For later markers, sum over every possible preceding state and multiply by the current emission:

The sum of the final forward probabilities is the probability of the observed marker sequence under the model. To obtain a posterior at an earlier marker using all observations, define \(\beta_j(k)=P(X_{j+1}=x_{j+1},\ldots,X_m=x_m\mid Z_j=k)\). Then \(\beta_m(k)=1\), and working backward gives \(\beta_j(k)=\sum_l P(Z_{j+1}=l\mid Z_j=k)P(X_{j+1}=x_{j+1}\mid Z_{j+1}=l)\beta_{j+1}(l)\). Combining the two parts yields

The numerator is the joint probability of the observations and state \(k\) at marker \(j\); the denominator sums that joint probability over all possible states at that marker. The ratio is therefore a conditional probability, not a direct observation of ancestry.

Following Evidence Across Several Markers

The forward recursion avoids listing every possible ancestry path. With \(K\) states and \(m\) markers, it updates a vector of \(K\) probabilities at each marker, summing over the \(K\) possible preceding states. The following example shows how evidence can shift as markers accumulate. For simplicity, it uses two states, equal ancestry proportions, a transition probability of 0.90 to stay in the same state and 0.10 to switch, and allele frequencies \(f_A=0.80\) and \(f_B=0.20\) at each marker. These are hypothetical values.

Interpreting Local Ancestry Probabilities

A posterior such as 0.90 for A at a marker means that, conditional on the observed marker data and the specified HMM inputs and assumptions, the model assigns probability 0.90 to state A there. It is not a 90% probability that every base in a surrounding region has ancestry A. Nor does it account automatically for uncertainty in allele-frequency estimates, admixture history, phasing, or the choice of model. Those limitations matter when probabilities are used to define genomic segments.

A local ancestry analysis commonly reports a probability for each ancestry at each marker, or summarizes nearby markers into segments. A segment label is a simplification of the underlying probabilities: boundaries can be uncertain, especially where markers provide weak evidence or where states compete closely. A threshold for assigning a label should therefore be understood as a reporting rule, not as proof that the ancestry changes exactly at that marker.

There is also a distinction between the most probable ancestry at each marker and the most probable sequence of states across the whole chromosome. Choosing the largest posterior separately at each marker gives pointwise labels. The most probable joint path instead maximizes the probability of the entire sequence of hidden states. These need not agree, because the latter accounts for transitions jointly across markers.

Common Mistakes

• Using the allele-1 frequency for an observed allele 0. If the source frequency is \(f_{kj}\), then \(P(X_j=0\mid Z_j=k)=1-f_{kj}\). Always match the emission to the observed allele.
• Reading a posterior as certainty. A high local ancestry probability is conditional on the model and data; it is not a direct observation or a guarantee that a whole segment has that ancestry.
• Ignoring distance between markers. The transition probability depends on genetic distance and the assumed history. Applying one transition matrix to intervals of very different lengths can misrepresent persistence.
• Confusing global proportions with local states. An ancestry proportion summarizes ancestry across a much larger scale; it does not specify the ancestry at an individual marker.
• Assuming nearby marker emissions are automatically independent. The simple HMM emission calculation makes a conditional-independence assumption. Strong residual dependence between markers can undermine that model.