Conditional Wright–Fisher Process

• Conditional Wright–Fisher process is a family of models where allele frequency dynamics are conditioned on endpoints, fixation, ancestry, or observation data.
• The approach incorporates mechanisms such as bridge diffusions, rescaled Pólya urn predictive means, and dual ancestral conditioning, modifying the classical drift and diffusion.
• These constructions enable exact simulation, filtering in hidden Markov settings, and maintain the Wright–Fisher covariance structure through conditional reweighting.

The conditional Wright–Fisher process is not a single canonical stochastic process but a family of Wright–Fisher-type constructions in which the law is conditioned, reweighted, projected, or inferred relative to additional information. In recent work, this includes endpoint-conditioned bridge diffusions, fixation-conditioned and ancestry-conditioned $\Lambda$-Wright–Fisher models, predictive-mean dynamics arising from rescaled Pólya urns, posterior signal laws in hidden Wright–Fisher systems, and pathwise constructions based on intertwining, excursion theory, or time reversal (Griffiths et al., 2017, Fittipaldi et al., 25 Nov 2025, Baake et al., 2016, Aletti et al., 2021, Boetti et al., 2024). What unifies these objects is that the state variable no longer represents only the unconditioned allele frequency; it may instead be a conditional expectation, a bridge path pinned at both endpoints, a frequency process under a change of measure, a conditional ancestor distribution, or a conditional law given observations.

1. Conceptual scope and main conditioning paradigms

A convenient one-dimensional reference form is the Wright–Fisher diffusion with mutation

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$

whose state space is $[0,1]$ and whose diffusion coefficient degenerates at the boundary (Jenkins et al., 2023). In multidimensional settings, the covariance retains the Wright–Fisher form $x_i(\delta_{ij}-x_j)$, while conditioning modifies either the drift, the path law, the latent representation, or the interpretation of the state.

The major conditioning mechanisms appearing in the literature can be organized as follows.

Mechanism	Conditioned object	Reference
Endpoint conditioning	Bridge $X(t)\mid X(0)=x,\ X(T)=z$	(Griffiths et al., 2017, Sant et al., 2023)
Predictive conditioning	Predictive mean $E[\xi_{n+1}\mid\mathcal F_n]$	(Aletti et al., 2021)
Fixation conditioning	Process conditioned on eventual fixation	(Fittipaldi et al., 25 Nov 2025)
Ancestral conditioning	$h(x)=\mathbb P(I_0=0\mid X_0=x)$	(Baake et al., 2016)
Observation conditioning	Filtering and smoothing laws	(Boetti et al., 2024)
Auxiliary pathwise conditioning	Intertwining, excursions, time reversal	(Hudec, 2017, Jenkins et al., 2023, Pal, 2010)

This taxonomy is important because identical terminology can hide mathematically distinct constructions. In bridge theory, conditioning is imposed on a future endpoint and is typically realized by a Doob-type density ratio. In fixation-conditioned models, the entire forward law is altered by a change of measure. In predictive formulations, the conditioned object is itself a conditional expectation rather than the raw sampling process. In hidden-Markov formulations, the “conditional process” is a posterior law given data rather than a new population-genetic diffusion in isolation.

A recurrent source of confusion is the assumption that every conditional Wright–Fisher object is a diffusion on the original simplex with a modified drift. The cited literature shows that this is false. Some constructions enlarge the state space with latent variables, some produce mixtures of kernels rather than a closed-form SDE, and some reinterpret the same unconditioned diffusion through conditional path decompositions.

2. Predictive conditioning via rescaled Pólya urns

A particularly explicit conditional interpretation appears in the rescaled Pólya urn model, where the state is the predictive probability of the next draw rather than the empirical frequency itself. For a $k$-color urn with

$$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$

the predictive probabilities are

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$

The explicit representation

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$0

shows that the contribution of $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$1 at time $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$2 is $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$3, so for $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$4 the reinforcement is local and recent observations dominate (Aletti et al., 2021).

The predictive recursion can be written as

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$5

with

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$6

This is a stochastic approximation scheme with a deterministic pull toward $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$7 and a martingale noise term. Under the normalization $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$8, the total mass becomes constant, and after the diffusive rescaling

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$9

the predictive process converges weakly in $[0,1]$0 to the $[0,1]$1-alleles Wright–Fisher diffusion with mutation

$[0,1]$2

with covariance

$[0,1]$3

This is the most literal use of the phrase “conditional Wright–Fisher process” in the supplied literature. The limiting diffusion is not the scaling limit of the raw draw sequence $[0,1]$4, but of the conditional law of the next observation, encoded by $[0,1]$5. Mutation enters through the drift toward $[0,1]$6, while reinforcement generates the quadratic variation $[0,1]$7. The process is therefore conditional in the precise sense that its state variable is itself a conditional expectation.

Several structural consequences follow. Grouping colors into blocks preserves the limiting Wright–Fisher form with aggregated mutation kernel $[0,1]$8. Any grouped two-component process, and any single coordinate $[0,1]$9, satisfies a one-dimensional Wright–Fisher-type SDE on $x_i(\delta_{ij}-x_j)$0 with drift toward $x_i(\delta_{ij}-x_j)$1 and diffusion coefficient $x_i(\delta_{ij}-x_j)$2. The invariant density on the simplex is Dirichlet,

$x_i(\delta_{ij}-x_j)$3

that is, $x_i(\delta_{ij}-x_j)$4. Probabilistically, the near-critical regime $x_i(\delta_{ij}-x_j)$5 is exactly the regime in which drift and noise balance on the scale $x_i(\delta_{ij}-x_j)$6, producing a Wright–Fisher limit.

3. Endpoint-conditioned diffusions and bridge laws

In classical bridge theory, the conditional Wright–Fisher process is a Wright–Fisher diffusion conditioned to hit a prescribed endpoint at a prescribed time. If $x_i(\delta_{ij}-x_j)$7 is the underlying diffusion, the bridge from $x_i(\delta_{ij}-x_j)$8 at time $x_i(\delta_{ij}-x_j)$9 to $X(t)\mid X(0)=x,\ X(T)=z$0 at time $X(t)\mid X(0)=x,\ X(T)=z$1 is

$X(t)\mid X(0)=x,\ X(T)=z$2

with transition density

$X(t)\mid X(0)=x,\ X(T)=z$3

This is the standard Doob-conditioning formula, and it makes explicit that the bridge is the original diffusion reweighted by a future endpoint factor rather than a process determined only by local drift and diffusion coefficients (Griffiths et al., 2017).

The most studied singular regime is the neutral $X(t)\mid X(0)=x,\ X(T)=z$4 bridge, where

$X(t)\mid X(0)=x,\ X(T)=z$5

This models an allele absent at the beginning and end of the observation window but present at intermediate times. In the neutral no-mutation case, the bridge density at time $X(t)\mid X(0)=x,\ X(T)=z$6 is

$X(t)\mid X(0)=x,\ X(T)=z$7

where

$X(t)\mid X(0)=x,\ X(T)=z$8

The genealogical interpretation is that, from an intermediate time $X(t)\mid X(0)=x,\ X(T)=z$9, lineages coalesce backward toward time $E[\xi_{n+1}\mid\mathcal F_n]$0 and also coalesce toward time $E[\xi_{n+1}\mid\mathcal F_n]$1; the bridge genealogy is therefore two-sided rather than a single backward tree. In the presence of selection, the genealogy becomes branching-coalescing rather than purely coalescent, but the bi-directional structure remains (Griffiths et al., 2017).

Bridge conditioning has also acquired an exact simulation theory. For neutral and selected Wright–Fisher bridges, EWF represents bridge densities as mixtures of Beta distributions with latent indices arising from coalescent-count variables and binomial counts. In the two-allele model with mutation $E[\xi_{n+1}\mid\mathcal F_n]$2, selection intensity $E[\xi_{n+1}\mid\mathcal F_n]$3, and polynomial selection function $E[\xi_{n+1}\mid\mathcal F_n]$4, the underlying SDE is

$E[\xi_{n+1}\mid\mathcal F_n]$5

The bridge law at an intermediate time $E[\xi_{n+1}\mid\mathcal F_n]$6 is again

$E[\xi_{n+1}\mid\mathcal F_n]$7

and the latent mixture is sampled exactly via alternating-series methods and Beta draws (Sant et al., 2023).

A different but related endpoint-conditioned viewpoint appears in the path-integral formulation of the selected Wright–Fisher transition density. There the selected density $E[\xi_{n+1}\mid\mathcal F_n]$8 is written as an integral over all neutral Wright–Fisher paths with fixed endpoints $E[\xi_{n+1}\mid\mathcal F_n]$9 and $h(x)=\mathbb P(I_0=0\mid X_0=x)$0, weighted by a Girsanov factor. The integration domain is therefore inherently endpoint-conditioned, but the work explicitly does not develop a separate conditional Wright–Fisher theory in the probabilistic sense of a bridge or a Doob-transformed process (Schraiber, 2013).

4. Fixation, ancestry, and dual conditioned structures

Conditioning can also target genealogical outcomes rather than path endpoints. In the two-type $h(x)=\mathbb P(I_0=0\mid X_0=x)$1-Wright–Fisher process with selection and mutation, a central quantity is the common ancestor type distribution

$h(x)=\mathbb P(I_0=0\mid X_0=x)$2

where $h(x)=\mathbb P(I_0=0\mid X_0=x)$3 is the current beneficial-type frequency and $h(x)=\mathbb P(I_0=0\mid X_0=x)$4 is the type of the immortal line in the stationary ancestral graph. The process is governed by a generator with a $h(x)=\mathbb P(I_0=0\mid X_0=x)$5-reproduction integral term, a Kingman diffusion term $h(x)=\mathbb P(I_0=0\mid X_0=x)$6, and a selection-mutation drift $h(x)=\mathbb P(I_0=0\mid X_0=x)$7. The representation

$h(x)=\mathbb P(I_0=0\mid X_0=x)$8

expresses the conditional ancestor type distribution as a Bernstein-type series whose coefficients are equilibrium tail probabilities of the pruned lookdown line-counting process $h(x)=\mathbb P(I_0=0\mid X_0=x)$9 (Baake et al., 2016).

The same paper identifies a strong pathwise Siegmund dual $k$0 of $k$1, satisfying

$k$2

and proves that

$k$3

Thus the coefficients in the ancestor-type expansion are hitting probabilities of a dual process. This is a conditional Wright–Fisher structure in which the conditioned object is not the forward allele frequency itself but the type of the eventual common ancestor given the present frequency.

A stronger forward conditioning occurs in the $k$4-Seed-Bank-Wright–Fisher process conditioned on fixation of a specified type $k$5. The fixation event is

$k$6

After a change of measure, the conditioned process is again a seed-bank Wright–Fisher process, but now with coordinated mutations governed by a random switching environment $k$7 with rates $k$8 and $k$9. The mutation measure satisfies

$$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$0

The conditioned process is the unique strong solution of the displayed SDE system in the paper, with extra drift and jump terms in the $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$1-equation driven by $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$2 and the compensated Poisson measure $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$3 (Fittipaldi et al., 25 Nov 2025).

The lookdown interpretation is especially sharp: fixation of $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$4 occurs if and only if the lowest level is of type $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$5 at time $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$6. Under this conditioning, the first level becomes the environment process, and reproduction events involving that level are reinterpreted as coordinated mutations for the rest of the population. The genealogy of the conditioned process becomes a structured $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$7-coalescent with seed-bank switching and coordinated mutations.

Related unconditioned $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$8-Wright–Fisher results clarify why these genealogical conditionings are tractable. In the multi-type setting, fixation, coupling, and stationary times are represented by explosion times of generalized fixation lines; with mutation, the explosion time $$N_{n,i}=b_i+B_{n,i},\qquad B_{n+1,i}=\beta B_{n,i}+\alpha \xi_{n+1,i},$$9 is a stationary time, and in the pure Kingman case it is a strong stationary time (Blancas et al., 2023). This does not by itself define a conditioned process, but it supplies the line-counting and lookdown infrastructure on which conditioning by fixation or stationarity naturally rests.

5. Observation-conditioned Wright–Fisher diffusions

A distinct meaning of conditional Wright–Fisher process arises in hidden-Markov models, where the diffusion is an unobserved signal and one seeks its conditional law given discrete-time samples. For coupled Wright–Fisher diffusions at $$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$0 loci, the signal solves

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$1

with within-locus covariance

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$2

parent-independent mutation

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$3

and cross-locus coupling through the quadratic fitness potential

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$4

The diffusion is reversible with stationary law

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$5

where $$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$6 is the product Dirichlet kernel (Boetti et al., 2024).

Observations are multinomial: $$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$7 The central result is that filtering and smoothing distributions remain in a conjugate family of countable mixtures of tilted Dirichlet kernels. If

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$8

then prediction gives

$$\psi_n=E[\xi_{n+1}\mid\mathcal F_n]=\frac{N_n}{|N_n|}=\frac{b+B_n}{r_n^*}, \qquad r_n^*=|N_n|.$$9

and the next update is

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$00

The weights are determined by the transition probabilities of a countable-state dual jump process $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$01, together with multinomial-Dirichlet integrals $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$02. The duality relation

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$03

is the structural reason that conditional distributions can be updated sequentially despite the absence of a closed-form diffusion semigroup. In this setting, the conditional Wright–Fisher process is a posterior law, not a changed forward dynamic. The process remains unconditioned in the population-genetic sense, but inference is entirely about its conditional distribution given data.

This perspective broadens the notion of conditioning in Wright–Fisher theory. Instead of conditioning on a future boundary event or ancestral outcome, one conditions on an observation history, and the resulting object is a dynamically updated mixture on the simplex. The technical content is correspondingly different: reversibility, duality, and conjugacy replace Doob transforms and lookdown changes of measure.

6. Auxiliary pathwise representations and conceptual boundaries

Several other constructions provide conditional or quasi-conditional Wright–Fisher viewpoints without fitting the bridge, fixation, or filtering templates exactly. One is intertwining. For the Wright–Fisher diffusion reflected at $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$04 and absorbed at $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$05,

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$06

Hudec constructs a pure birth process $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$07 with rates

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$08

and a kernel

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$09

The intertwining identity

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$10

implies the conditional law

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$11

and the absorption time of $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$12 equals the explosion time of $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$13 almost surely (Hudec, 2017). This is a bona fide conditional statement, but not a bridge or a Doob-conditioned diffusion on the original state space; it is a coupled Markov pair.

Excursion theory produces another conditional decomposition. For the Wright–Fisher diffusion with mutation parameters $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$14, killed upon hitting either endpoint, excursions start from one boundary and end at one of the two boundaries. The excursion process is a marked Poisson point process $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$15 with intensity $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$16, indexed by boundary local time. Entrance laws $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$17, killed semigroups, and explicit hypergeometric hitting-time transforms describe the conditional motion between boundary visits, and concatenating these killed excursions reconstructs the full diffusion path (Jenkins et al., 2023). Here the conditional Wright–Fisher object is a path segment conditioned on its boundary origin and boundary destination.

Time reversal provides a further variant. The Wright–Fisher diffusion with negative mutation rates on the simplex,

$$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$18

is killed at the boundary and can be viewed as the stochastic time reversal of a classical Wright–Fisher process of increasing dimensions conditioned at a random time (Pal, 2010). The conditioning is on terminal values of the normalized positive-dimension process at that random time. This result shows that conditioned Wright–Fisher dynamics may emerge from reversing an enlarged unconditioned system rather than modifying the forward generator directly.

A related geometric interpretation appears in the hierarchical extension of the forward Wright–Fisher equation on the simplex. Probability flux leaving a face is recursively turned into lower-dimensional evolution on the next stratum, so allele loss is treated as continuation rather than termination. The paper explicitly states that, in conditional Wright–Fisher language, this can be read as a conditioned-and-continued dynamics on lower-dimensional faces after extinction events (Hofrichter et al., 2014).

These constructions also delimit the concept. Some highly relevant papers do not define a conditional Wright–Fisher process in a strict sense. The path-integral approach to selection is endpoint-conditioned at the level of path integration, but it does not introduce a separate conditioned process (Schraiber, 2013). The multidimensional $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$19-Wright–Fisher framework with general frequency-dependent selection supplies fixation probabilities, ancestral structure, and extinction theory, but explicitly does not provide a Doob $$\mathscr{G} =\frac12 x(1-x)\frac{d^2}{dx^2} +\frac12\bigl(\theta_1-(\theta_1+\theta_2)x\bigr)\frac{d}{dx},$$20-transform or formal conditioned-process construction (Casanova et al., 2019). The terminology is therefore best understood as a family resemblance rather than a universally standardized definition.

Across these variants, the conditional Wright–Fisher process serves as a unifying theme for several different mathematical operations: conditioning on endpoints, conditioning on fixation, conditioning on ancestry, conditioning on observations, and extracting hidden conditional dynamics from predictive means or coupled latent processes. The common outcome is that the classical Wright–Fisher covariance structure survives, while the state interpretation and the mechanism generating the drift or path law become conditional.