# Inference in molecular population genetics

**Date**: 03/08/2026
**Domain**: statistics/computational_statistics
**Taxonomy**: academic/research_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Reliability of Variance Estimates and Standard Errors**

Multiple reviewers note that the finiteness of the importance weight variance has not been proven for general models, which challenges the validity of the Central Limit Theorem justification used for standard errors. Empirical results in Table 1 suggest standard errors may severely underestimate true uncertainty when weight distributions are skewed, such as at theta equals 15.0. It would be helpful to apply the extreme value theory diagnostics mentioned in Section 6.4 to the main results or clearly label standard error estimates as heuristic where variance finiteness is unestablished.

**Sensitivity to the Driving Value Theta_0**

The methodology relies on a single driving value theta_0 for the proposal distribution, yet the sensitivity of likelihood estimates to deviations from this value is not quantitatively bounded. While Section 6.1 acknowledges that likelihood may be underestimated away from theta_0, readers might note that the robustness of the likelihood curves in Figures 2 through 5 depends heavily on this choice. It would be beneficial to include a systematic analysis of estimation accuracy as a function of the distance from theta_0 or to incorporate multi-theta_0 bridge sampling strategies into the primary methodology.

**Clarification of Efficiency and Scalability Claims**

The abstract claims efficiency improvements of several orders of magnitude, but computational times for the NSE dataset reached 18 hours per run, which limits practical utility for larger problems. There is also a need to distinguish between statistical efficiency and computational cost per iteration, as the proposed proposal distribution can be computationally heavier than alternatives. It would be helpful to temper scalability claims with caveats regarding problem size and to provide clearer criteria for when MCMC might be preferred over the proposed importance sampling method.

**Quantification of Approximation Errors and Theoretical Extensions**

The implementation relies on Gaussian quadrature and approximations of conditional probabilities that are described as rough in some cases, without quantified bounds on how these errors propagate to likelihood estimates. Additionally, the extension to the infinite sites model is presented as an analogy to the finite case theory, suggesting theoretical guarantees may not strictly apply. It would be helpful to include sensitivity analyses showing the relationship between quadrature precision and weight variance, and to clarify the theoretical status of the infinite sites adaptation.

**Robustness to Model Assumption Violations**

The current framework assumes constant population size, no recombination, and specific mutation models, yet real genetic data often violate these assumptions. The authors acknowledge that little is known about the robustness of likelihood procedures under such violations, but the results are presented as if assumptions hold. It would be helpful to add a sensitivity analysis showing how likelihood estimates change when data are generated under model violations, or to clearly frame the results as conditional on model correctness.

**Panmixia Assumption vs. Multi-Population NSE Data**

The coalescent model formally assumes a single panmictic population with exchangeable lineages (Section 2). However, the NSE data set analyzed in Section 5.4 combines 60 males from three geographically distinct populations (Nigeria, Sardinia, and East Anglia). These populations have different demographic histories and limited gene flow, violating the exchangeability assumption. The consequence is that likelihood estimates for θ may be biased, as the model attributes between-population variation to mutation rather than population structure. The authors acknowledge in Section 6.2 that extensions to structured populations exist (Bahlo and Griffiths, 2000) but do not apply them here. A concrete fix would be to either analyze each population separately or implement the structured coalescent extension mentioned.

**Stationarity Assumption vs. Human Demographic History**

The model assumes samples are 'taken from the population at stationarity' (Section 2), implying constant population size over evolutionary time. However, human populations (including the Y-chromosome data used in Section 5.4) are known to have experienced recent expansions and bottlenecks, as noted by discussant Harding. This violates the stationarity assumption underlying the coalescent approximation. The consequence is that θ estimates conflate mutation rate with population size changes, potentially biasing ancestral inference. The authors acknowledge in Section 6.2 that extensions to varying population size exist (Griffiths and Tavaré, 1994c) but proceed with the constant-size model. Readers might note that applying the method to human data without addressing demographic non-stationarity limits the biological interpretability of results.

**Status**: [Pending]

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## Detailed Comments (21)

### 1. Stationary Distribution Assumption Scope

**Status**: [Pending]

**Quote**:
> We assume that the evolution is neutral, in that the demography is independent of the genetic types of the chromosomes. To simplify the description we focus on the case where $P$ has a unique stationary distribution.

**Feedback**:
The section states that the description focuses on cases where the mutation matrix $P$ has a unique stationary distribution. However, later in the same section, the text discusses the infinite sites model where $P$ does not have a stationary distribution. It would be helpful to clarify that the stationary distribution assumption applies primarily to finite allele models to prevent contradiction with the Infinite Sites model discussion.

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### 2. Notation Collision for Sample Size n

**Status**: [Pending]

**Quote**:
> where $n$ is the number of chromosomes in $H_{i-1}$ and $n_{\alpha}$ is the number of chromosomes of type $\alpha$ in $H_{i-1}$.

**Feedback**:
The symbol $n$ is globally defined earlier in the section as the fixed size of the random sample $A_n$. However, in Equation (1) and the accompanying text, $n$ is redefined locally as the variable number of chromosomes in the intermediate state $H_{i-1}$. Readers might confuse the fixed parameter $n$ with the variable state size when implementing the transition kernel, so using a distinct symbol like $k$ for the variable state size would improve clarity.

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### 3. Combinatorial formula mislabeled as topologies

**Status**: [Pending]

**Quote**:
> For example, there are $n!(n-1)!/2^{n-1}$ different possible topologies for the underlying trees.

**Feedback**:
The formula $n!(n-1)!/2^{n-1}$ calculates the number of ranked labeled histories (coalescent sequences), not the number of topologies. Using the term 'topologies' here misrepresents the combinatorial complexity being described to justify the computational challenge. It would be helpful to use the term 'ranked labeled histories' to accurately reflect the size of the integration space $\mathcal{H}$.

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### 4. Zero probability claim inaccurate for finite-alleles models

**Status**: [Pending]

**Quote**:
> Since $\pi_{\theta}(A_n | \mathcal{H})$ is 0 unless $\mathcal{H}$ is consistent with $A_n$, each term of the sum will be 0 with very high probability

**Feedback**:
This claim holds for infinite sites models where incompatibility leads to zero probability, but it is not accurate for finite-alleles models like Parent-Independent Mutation (PIM) discussed later. In PIM, any genealogy can generate any data configuration with non-zero probability, so the terms are small but not strictly zero. It would be helpful to distinguish between models where consistency is binary versus probabilistic to ensure the motivation for importance sampling applies generally.

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### 5. Misattribution of single-chain likelihood estimation capability

**Status**: [Pending]

**Quote**:
> Griffiths and Tavaré (1994a) showed how to use realizations of a single Markov chain to estimate the likelihood function.

**Feedback**:
This claim appears inconsistent with Section 4's motivation and the standard understanding that GT 1994a requires $\theta$-specific simulation. If GT 1994a already enabled single-chain estimation for the likelihood function, the novelty of the current paper's IS scheme is obscured. It would be helpful to clarify that GT 1994a proposed the recursion/MC method, while the efficient single-chain IS estimation is the contribution of the current work.

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### 6. Product limit in likelihood expectation equation

**Status**: [Pending]

**Quote**:
> he form
> 
> $$
> L(\theta) = \pi_{\theta}(A_n) = E \left\{ \prod_{j=0}^{\tau} F(B_j) \,\Big| \, B_0 = A_n \right\} \tag{6}
> $$
> 
> where $B_0, B_1, \ldots, B_\tau$ is a partic

**Feedback**:
The product index $j=0$ to $\tau$ includes $\tau+1$ terms, but the chain $B_0 \to \dots \to B_\tau$ has only $\tau$ transitions. In importance sampling estimators, weights are typically products over transitions. It would be helpful to rewrite the product limit as $\tau-1$ or define $F(B_\tau)=1$ explicitly to ensure mathematical consistency with the number of transitions.

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### 7. MCMC autocorrelation not addressed in variance calculation

**Status**: [Pending]

**Quote**:
> If $\mathcal{T}^{(1)}, \mathcal{T}^{(2)}, \ldots$ is a sample (after burn-in, and possibly thinning) from a Markov chain with stationary distribution $P_{\theta_0}(\mathcal{T} | A_n)$

**Feedback**:
The text acknowledges samples come from a Markov chain and mentions thinning, but equation (7) treats these as independent draws in the Monte Carlo approximation. For correlated MCMC samples, the variance of the estimator depends on the autocorrelation structure, not just sample size M. It would be helpful to add 'accounting for autocorrelation in the chain' and cite a reference on MCMC standard error estimation because the current presentation implies independent sampling.

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### 8. MCMC method description conflates with importance sampling reweighting

**Status**: [Pending]

**Quote**:
> A natural alternative to the Griffiths–Tavaré scheme, which was first pursued by Kuhner et al. (1995, 1998), is to deal with the missing data by using MCMC methods.

**Feedback**:
The text states MCMC constructs a chain with stationary distribution $P_{\theta}(\mathcal{T} | A_n)$ for the target parameter, but equation (7) actually samples from $P_{\theta_0}(\mathcal{T} | A_n)$ at a fixed driving value and reweights. This is importance sampling applied to MCMC output, not pure MCMC inference at each $\theta$. It would be helpful to add a clarifying sentence stating 'However, rather than running separate chains for each $\theta$, we sample from one chain at $\theta_0$ and reweight' because the current presentation obscures the actual computational strategy.

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### 9. Inconsistent equation numbering for estimator

**Status**: [Pending]

**Quote**:
> the estimator (8) is asymptotically normal with variance $\sigma^2 / M$.

**Feedback**:
In Section 3.4, Equation (9) is explicitly defined as the Monte Carlo estimator (the sum), while Equation (8) is the exact integral identity. However, this statement in Section 5 attributes the properties of the estimator to the integral identity (8) instead of the estimator (9). Readers might note this inconsistency when cross-referencing the variance theorems in Section 5 with the definitions in Section 3, so rewriting the reference to cite 'estimator (9)' would align with the established definitions.

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### 10. Proposed solution does not satisfy recurrence in Proposition 1(d)

**Status**: [Pending]

**Quote**:
> If $\tilde{\pi}$ satisfies property (b) of the proposition, then $M^{(1)} = (1 - \lambda_1)(I - \lambda_1P)^{-1}$.

**Feedback**:
The claimed solution $M^{(n)} = (1 - \lambda_n)(I - \lambda_n P)^{-1}$ is stated to follow from the recurrence in equation (23), but algebraic verification suggests potential discrepancies for general $P$ and $\theta > 0$. Readers might note that the recurrence requires specific relationships between $M^{(n)}$ and $M^{(n+1)}$ that should be explicitly verified. It would be helpful to provide the intermediate algebraic steps or confirm the closed-form solution satisfies the recurrence to ensure the proof is robust.

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### 11. Technical imprecision in variance underestimation claim

**Status**: [Pending]

**Quote**:
> However, caution is necessary, as even assuming finite variance (which is not guaranteed) the distribution of the weights may be so highly skewed that, even for very large $M$, $\sigma^2$ is (with high probability) underestimated by $\hat{\sigma}^2$

**Feedback**:
The claim that $\sigma^2$ is underestimated 'with high probability' by the sample variance $\hat{\sigma}^2$ is technically imprecise, as the sample variance is an unbiased estimator for i.i.d. samples regardless of skewness. What actually occurs with heavy-tailed IS weight distributions is that $\hat{\sigma}^2$ has high variability and converges slowly. It would be helpful to rewrite this to state that variance estimates are unreliable due to high variability rather than systematic underestimation.

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### 12. Logical error in boundary condition description

**Status**: [Pending]

**Quote**:
> truncating the type space $E$ to $\{0, 1, \ldots, 19\}$ by insisting that all mutations to alleles of length 0 or 19 involve the gain or loss respectively of a single repeat.

**Feedback**:
The description of the boundary condition contains a logical impossibility, as a mutation resulting to an allele of length 0 must involve a loss of a repeat (from length 1), not a gain as stated. The text likely intends to describe reflecting boundaries where mutations from alleles of length 0 involve a gain. It would be helpful to rewrite 'mutations to alleles' as 'mutations from alleles' to correctly describe the reflecting boundary behavior intended for the truncated type space.

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### 13. Inconsistent parameter notation in NSE analysis

**Status**: [Pending]

**Quote**:
> Five different estimated likelihood and relative likelihood curves were found by using our method, each based on half a million samples from $Q_{\pi_0=8.0}^{\mathrm{SD}}$.

**Feedback**:
The notation $Q_{\pi_0=8.0}^{\mathrm{SD}}$ uses $\pi_0$ to denote the driving value parameter, which contradicts the standard notation $\theta_0$ used consistently elsewhere in the section and Section 4 definitions. This inconsistency creates ambiguity about whether a different parameter or distribution is being referenced for the NSE data analysis. It would be helpful to rewrite $Q_{\pi_0=8.0}^{\mathrm{SD}}$ as $Q_{\theta_0=8.0}^{\mathrm{SD}}$ to maintain notational consistency with the rest of the paper.

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### 14. Equation (30) uses incorrect notation for likelihood term

**Status**: [Pending]

**Quote**:
> interspecies comparisons can be performed which typically provide a substantial amount of information on the type of the MRCA. Using IS as before gives
> 
> $$
> P _ {\theta} \left(A _ {n}, O\right) \approx \frac {1}{M} \sum_ {i = 1} ^ {M} \pi_ {\theta} \left(A _ {n} \mid \mathcal {H} ^ {(i)}\right) P _ {\theta} \left(O \mid \mathcal {H} ^ {(i)}\right) \frac {P _ {\theta} \left(\mathcal {H} ^ {(i)}\right)}{Q _ {\theta} \left(\mathcal {H} ^ {(i)}\right)}. \tag {30}
> $$
> 
> T

**Feedback**:
The term $\pi_{\theta}(A_n | \mathcal{H}^{(i)})$ uses notation inconsistent with the definition of $\pi$ from Section 4, where $\pi$ takes a type as the first argument and a sample configuration as the second. In the Stephens-Donnelly IS framework, histories are sampled backwards from the data and are consistent with $A_n$ by construction, making $P_{\theta}(A_n | \mathcal{H}^{(i)}) = 1$. It would be helpful to rewrite equation (30) to remove this term or use $P_{\theta}(A_n | \mathcal{H}^{(i)})$ if clarity is needed, as histories are sampled consistent with $A_n$ by construction.

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### 15. Likelihood Linear System Claim Inaccurate For Infinite Sites

**Status**: [Pending]

**Quote**:
> Since the likelihood in question is the solution of a system of linear equations, it may also be fruitful to pursue other (non-stochastic) numerical methods.

**Feedback**:
The claim that the likelihood is the solution of a system of linear equations is mathematically precise for finite-allele models, but this section also discusses the infinite sites model, which has an unbounded state space and is not described by a finite linear system. This creates an internal inconsistency within the section regarding the mathematical characterization of the likelihood across the different models discussed. It would be helpful to qualify this statement to specify that it applies to finite-allele models or recurrence-based frameworks.

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### 16. Ascertainment modeling description lacks precision

**Status**: [Pending]

**Quote**:
> adapting both $P_{\theta}(\mathcal{H})$ and $Q_{\theta}(\mathcal{H})$ so that mutations can occur only on panel lineages.

**Feedback**:
The statement 'mutations can occur only on panel lineages' is imprecise and could mislead readers, as biologically mutations occur on all lineages in the genealogy. What the authors mean is that for the likelihood calculation under this ascertainment scheme, only mutations on panel lineages are observable. It would be helpful to rewrite this to clarify that the modification is to the likelihood calculation, not the biological process, by stating 'conditioning on mutations being observable only on panel lineages'.

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### 17. Clarify definition of finite histories in infinite sites model

**Status**: [Pending]

**Quote**:
> except in the special case of the infinite sites model where the number of possible histories is finite.

**Feedback**:
Readers might note that in standard coalescent theory, a 'history' typically includes continuous coalescence times, making the set of possible histories uncountably infinite even under the infinite sites model. The authors likely refer to the discrete sequence of topological events, which is indeed finite due to the no-back-mutation constraint. It would be helpful to specify 'discrete event sequences' or 'topological histories' to align with the algorithm's actual sampling space and clarify the source of the variance finiteness.

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### 18. Routine use claim conflicts with documented bias

**Status**: [Pending]

**Quote**:
> methods described in the paper are becoming routinely used by biologists for much larger data sets, and in some more complex settings.

**Feedback**:
This statement appears to conflict with findings in Section 6.1 and 5.2, which indicate that the methods tend to underestimate the relative likelihood away from the driving value and can produce severely underestimated standard errors. Describing the methods as 'routinely used' without qualifying these known reliability issues might lead readers to overestimate the robustness of the inference. It would be helpful to acknowledge that while adoption is growing, the theoretical limitations regarding bias and variance identified in Sections 5.2 and 6.1 remain relevant constraints.

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### 19. Inconsistent Mutation Count Distributions

**Status**: [Pending]

**Quote**:
> where $m$ is geometrically distributed with parameter $l\theta / (n+l\theta)$.

**Feedback**:
Readers might note a discrepancy in the expected number of mutations between the two described processes in Appendix A. Under the standard definition of a geometric distribution with parameter $p$, the mean is $(1-p)/p$; using the stated parameter yields a mean of $n/l\theta$, but the equivalent Poisson process description implies a mean of $l\theta/n$. It would be helpful to clarify the definition of the geometric parameter to ensure implementation matches the theoretical derivation, possibly by using parameter $n / (n+l\theta)$.

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### 20. Support Condition for IS Consistency

**Status**: [Pending]

**Quote**:
> We used $s = 4$ quadrature points. Although in some cases the approximation to $\hat{\pi}(\cdot|\cdot)$ obtained through this procedure is rather rough, we note that in any case the IS function defined by this approximation to $\hat{\pi}(\cdot|\cdot)$ is a valid IS function in its own right, and so leads to an estimator (9) which is consistent.

**Feedback**:
The claim that the estimator is consistent relies on the importance sampling proposal having support everywhere the target distribution is non-zero. However, approximating the integral with a finite quadrature sum and truncating the matrix power series could result in zero probability estimates for reachable states. It would be helpful to add 'provided the approximation maintains non-zero support for all reachable states' to qualify the consistency claim.

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### 21. Felsenstein's sample size claim contradicts authors' extension

**Status**: [Pending]

**Quote**:
> ions. Stephens and Donnelly also point out that the variance of our importance sampling weights can become infinite when the trial value of $\theta$ used in the sample is smaller than the true value by more than a factor of 2. It is important to note that this rule is for $n = 2$. For more reasonable sample sizes the conditions for misbehaviour may be more stringent. For example, Fig. 11 shows th

**Feedback**:
Felsenstein claims the factor-of-2 variance condition applies specifically to n=2, suggesting larger sample sizes have more stringent conditions. However, the authors directly contradict this in their reply, stating the variance is infinite for $\theta > 2\theta_0$ regardless of the value of the sample size n. It would be helpful to rewrite Felsenstein's statement to reflect the authors' clarification that the condition holds regardless of sample size n, as this is a concrete mathematical disagreement in the text.

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