# Inference in molecular population genetics

**Date**: 03/07/2026
**Domain**: statistics/statistical_inference
**Taxonomy**: academic/review_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Reliability of Variance Estimates for Skewed Importance Weights**

Reviewers note that standard errors reported in the main results may underestimate true uncertainty due to skewed weight distributions, particularly in cases like \theta=15. While the authors suggest using generalized Pareto distributions to diagnose this, this procedure is not applied to the problematic examples. It would be helpful to integrate this diagnostic into the reported examples to demonstrate its effectiveness in detecting underestimation of uncertainty. Without this, readers might note that the reported precision could be misleadingly small.

**Sensitivity of Estimator Efficiency to Driving Value Selection**

The importance sampling estimator relies on a proposal distribution centered at a driving value \theta_0, yet the paper lacks a systematic method for selecting this value or analyzing how variance scales with the distance from the true parameter. Section 6.1 acknowledges that efficiency may degrade as \theta moves away from \theta_0, and readers might note that the variance of the estimator can become infinite if the driving value is too far. It would be helpful to include a simulation study quantifying this sensitivity to guide practitioners on when to re-run with a new driving value.

**Theoretical Consistency of the Proposal Distribution for Infinite Sites**

The extension to the infinite sites model is described as being made "by analogy" despite the mutation process being non-reversible, whereas the core theoretical results rely on reversibility. The lack of a rigorous derivation for this specific context leaves the efficiency of the proposal distribution unestablished for a primary application area. It would be helpful to clarify the extent to which the lack of reversibility affects the variance of the estimator or to provide a bound on the approximation error in this specific context.

**Scope Limitations Regarding Recombination and Demographic Assumptions**

The core methodology assumes constant population size and ignores recombination, which significantly limits the scope of the inference framework for modern genomic datasets. While extensions are mentioned in the discussion, the empirical validation is confined to these simplified models without assessing robustness to misspecification. It would be helpful to include a brief discussion or preliminary results on how the proposed strategy generalizes to the ancestral recombination graph or varying population sizes.

**Computational Overhead and Approximation Accuracy**

The paper claims substantial efficiency gains but does not explicitly account for the increased cost of calculating proposal weights compared to standard schemes, particularly when using Gaussian quadrature for large type spaces. Readers might note that the efficiency gains depend heavily on the quality of the approximation, and a sensitivity analysis of the quadrature points would strengthen the validation. It would be helpful to provide a breakdown of computational time per iteration versus the number of iterations required to achieve a target accuracy.

**Population Structure in NSE Data Application**

Section 2 defines the model for a 'random sample of $n$ chromosomes, taken from the population at stationarity' (page 608), which implies a single panmictic population. However, Section 5.4 applies the method to the NSE data set, which consists of '60 males from Nigeria, Sardinia and East Anglia' (page 626). Treating this as a single sample from one population violates the assumption of random mating within the sample. While Section 6.2 acknowledges extensions to structured populations exist, the likelihood estimates in Section 5.4 are derived under the single-population assumption. It would be helpful to clarify how population subdivision might bias the estimated $	heta$ values in this specific application.

**Finite Variance Assumption for Standard Errors**

Section 5 states that the estimator is asymptotically normal 'assuming that the distribution of the weights has finite variance $σ^2$' (page 620). However, in Section 5.2, the authors observe that for $θ = 15.0$, 'the distribution of the importance weights is so skewed that... calculated standard errors can seriously underestimate the standard deviation' (page 622). Furthermore, Section 6.4 notes, 'We could not prove finiteness of the variance of our weights, except in the special case of the infinite sites model' (page 632). Readers might note that the reported standard errors in Table 1 for $θ = 15.0$ may not reliably reflect uncertainty given this empirical skewness. It would be helpful to discuss robustness checks for the variance assumption when reporting standard errors.

**Status**: [Pending]

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## Detailed Comments (22)

### 1. Notation Ambiguity in Equation (1) Definition

**Status**: [Pending]

**Quote**:
> where $n$ is the number of chromosomes in $H_{i-1}$

**Feedback**:
The symbol $n$ is previously defined as the fixed sample size. In Equation (1), it is locally redefined as the variable lineage count. This dual usage creates ambiguity that could lead to implementation errors. It would be helpful to rewrite 'where $n$ is the number of chromosomes in $H_{i-1}$' as 'where $k$ is the number of chromosomes in $H_{i-1}$' and replace all instances of $n$ in Equation (1) with $k$ to distinguish the variable count from the fixed sample size.

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### 2. Algorithm Stopping Condition Overshoot

**Status**: [Pending]

**Quote**:
> Otherwise go back to the last time at which there were $n$ lines in the ancestry and stop.

**Feedback**:
The algorithm specifies that sampling continues until $n+1$ lines exist, then reverts to the state with $n$ lines. Standard coalescent algorithms typically terminate exactly when $n$ lineages are reached. Readers might note that it is not clear how this overshoot procedure affects the distribution of the sampled history compared to the standard $n$-coalescent. It would be helpful to add a justification after Step 3 explaining why the overshoot to $n+1$ lines is required to ensure the correct distribution of the history.

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### 3. Terminology Mismatch for Tree Count Formula

**Status**: [Pending]

**Quote**:
> different possible topologies for the underlying trees.

**Feedback**:
The formula provided calculates the number of labeled histories (ranked trees), whereas the term 'topologies' typically refers to unranked branching structures. While the coalescent generates histories, using the term 'topologies' for this formula may confuse readers regarding the dimensionality of the space being discussed. It would be helpful to rewrite 'different possible topologies' as 'different possible labeled histories' to accurately reflect the formula provided.

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### 4. Missing Support Condition for Reuse of Samples

**Status**: [Pending]

**Quote**:
> a single IS function. This is in theory straightforward: for any fixed $\theta_0$ we have
> 
> $$
> L (\t

**Feedback**:
Readers might note that the validity of Equation (12) relies on the support of the proposal distribution $Q_{	heta_0}$ covering the support of the target integrand for the specific $	heta$ being estimated. The text claims the approach holds for 'any fixed $	heta_0$' without explicitly stating this necessary condition. It would be helpful to rewrite 'This is in theory straightforward: for any fixed $	heta_0$ we have' as 'This is in theory straightforward: for any fixed $	heta_0$ whose support includes that of the target distribution for $	heta$, we have' to address the gap.

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### 5. Incorrect Count of Rooted Gene Trees

**Status**: [Pending]

**Quote**:
> (There are $S + 1$ such rooted gene trees, where $S$ is the number of mutations in the data.)

**Feedback**:
For an unrooted binary gene tree with $n$ sampled sequences, the number of edges is $2n - 3$. Since the root can be placed on any edge, there are $2n - 3$ possible rooted gene trees. The claim that there are $S + 1$ rooted trees is only true if $S = 2n - 4$. In general, $S$ can be less than $2n - 4$. Summing probabilities over only $S+1$ trees when $2n-3$ exist could lead to an underestimation of the likelihood. It would be helpful to rewrite the parenthetical statement to reflect the correct number of rooted trees ($2n-3$) or clarify the specific definition of 'rooted gene tree' used that restricts the count to $S+1$.

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### 6. Logical Error in Truncation Boundary Condition Description

**Status**: [Pending]

**Quote**:
> mutations to alleles of length 0 or 19

**Feedback**:
The text states the goal is truncating the type space to $	ext{0, …, 19}$. To enforce this, transitions *out* of the boundary states (0 and 19) must be restricted. The text describes restricting mutations *to* alleles of length 0 or 19 (i.e., transitions *into* the boundary). Restricting transitions into the boundary does not prevent the process from leaving the truncated space. It would be helpful to rewrite 'mutations to alleles of length 0 or 19' as 'mutations from alleles of length 0 or 19' because truncation requires restricting transitions leaving the boundary states.

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### 7. Inconsistent Notation for Driving Value Parameter

**Status**: [Pending]

**Quote**:
> ion samples from $Q_{\pi_0=8.0}^{\mathrm{SD}}$. The

**Feedback**:
The text uses $π_0$ here, but the driving value is consistently denoted as $θ_0$ throughout the section (e.g., in Fig. 3 and Fig. 5 captions, and earlier in the text). This creates ambiguity about whether $π_0$ refers to a different parameter or is a typo. It would be helpful to rewrite 'samples from $Q_{π_0=8.0}^{“SD”}$.' as 'samples from $Q_{θ_0=8.0}^{“SD”}$.' because the driving value parameter is consistently denoted as $θ_0$ elsewhere in the section.

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### 8. Normalization Error in Rao-Blackwellized Posterior

**Status**: [Pending]

**Quote**:
> $$ \pi(\theta|\mathcal{H}^{i}, A_{n}) = T^{i} \frac { (T^{i} \theta/2)^{m^{i}} \exp (-T^{i} \theta/2) }{ m^{i} ! } $$

**Feedback**:
Readers might note that the proposed posterior density for $θ$ given the genealogy does not integrate to 1. Integrating the right-hand side over $θ ∈ [0, ∞)$ yields 2. The normalization constant appears to be off by a factor of 2. It would be helpful to rewrite the equation as $$ \pi(\theta|\mathcal{H}^{i}, A_{n}) = \frac{(T^{i}/2)^{m^{i}+1}}{m^{i} !} \theta^{m^{i}} \exp (-T^{i} \theta/2) $$ to ensure it is a valid probability density.

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### 9. Coefficient Discrepancy in Griffiths-Tavaré Recursion

**Status**: [Pending]

**Quote**:
> $$ p(\mathbf{n}) = \frac{\theta}{n + \theta - 1} \left\{ \sum_{\alpha=1}^{d} \sum_{\beta=1}^{d} \frac{n_{\beta} + 1 - \delta_{\alpha \beta}}{n} P_{\beta \alpha} p(\mathbf{n} + \mathbf{e}_{\beta} - \mathbf{e}_{\alpha}) \right\} + \frac{n-1}{n + \theta - 1} \sum_{\{\alpha: n_{\alpha} > 0\}} \frac{n_{\alpha}-1}{n-1} p(\mathbf{n} - \mathbf{e}_{\alpha}) . \tag{34} $$

**Feedback**:
It would be helpful to verify the coefficients against the original Griffiths–Tavaré (1994) equation (2.1). The standard coefficient for the mutation term is $∑_{j} (n_j + δ_{ij})/n$ where $j$ is the child type and $i$ is the parent type. In the quoted text, with $β$ as child and $α$ as parent, the coefficient is given as $(n_{β} + 1 - δ_{α β})/n$. This inverts the delta term. It would be helpful to rewrite the coefficient as $(n_{β} + δ_{α β})/n$ to match the established recursion.

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### 10. Clarification on Likelihood Derivation from Conditionals

**Status**: [Pending]

**Quote**:
> rred to Section 6.
> 
> ## 5.1. Parent-independent mutation
> 
> For PIM models, the conditional distributions $\pi(\cdot|\cdot)$ are known exactly (see equation (16) above), and hence so also is the likelihood. 

**Feedback**:
The statement implies that knowledge of the conditional distributions $π(⋅|⋅)$ alone is sufficient to determine the likelihood. However, the likelihood $P(A_n)$ is the normalizing constant of the joint distribution $P(A_n, ℎ)$, and knowing the normalized conditional $P(ℎ|A_n)$ requires knowledge of the joint to recover the constant. It would be helpful to rewrite "and hence so also is the likelihood" as "and, given the known joint density, the likelihood is also known" to address this logical gap.

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### 11. Ambiguity in Quadrature Method Specification

**Status**: [Pending]

**Quote**:
> using Gaussian quadrature

**Feedback**:
The integral is over $[0, ∞)$ with weight $−	ext{exp}(-t)$, which specifically requires Gauss-Laguerre quadrature. The term 'Gaussian quadrature' is generic and could imply Gauss-Legendre on a finite interval. Specifying the quadrature type ensures reproducibility. It would be helpful to rewrite 'using Gaussian quadrature' as 'using Gauss-Laguerre quadrature' to specify the correct method for the infinite domain and exponential weight.

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### 12. Validity Claim for Approximate Proposal Distribution

**Status**: [Pending]

**Quote**:
> is a valid IS function in its own right

**Feedback**:
Consistency of the IS estimator requires the proposal distribution to be properly normalized. The text admits the approximation is 'rather rough' but asserts validity without demonstrating normalization. If the quadrature approximation does not sum to 1, the weights must be adjusted. It would be helpful to rewrite 'is a valid IS function' as 'is a valid proposal distribution, provided it is normalized' because consistency depends on the proposal being a proper probability distribution.

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### 13. Ambiguity in 'Finite Histories' Claim for 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**:
In standard coalescent theory, the number of genealogical histories (including continuous branch lengths) is infinite even under the Infinite Sites Model. The finiteness likely refers to the discrete ancestral configurations in the Griffiths-Tavaré algorithm rather than the histories themselves. This distinction is important because the variance finiteness depends on the boundedness of weights over the state space, not just the count of topologies. It would be helpful to rewrite 'except in the special case of the infinite sites model where the number of possible histories is finite' as 'except in the special case of the infinite sites model where the number of possible ancestral configurations is finite' to be technically precise.

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### 14. Mathematical Precision on Variance Estimator Consistency

**Status**: [Pending]

**Quote**:
> kewed that, even for very large $M$, $\sigma^2$ is (with high probability) underestimated by $\hat{\sigma}^2$, and/or the

**Feedback**:
For i.i.d. samples from a distribution with finite variance, the sample variance $σ^2$ is a consistent estimator of the true variance $σ^2$. By the Law of Large Numbers, $σ^2 → σ^2$ as $M → ∞$, meaning the probability of underestimation approaches 0.5, not 1. The caution regarding skewed weights is valid for finite $M$ or regarding the convergence of the mean's distribution, but the claim that $σ^2$ is underestimated with high probability for very large $M$ appears to contradict the consistency of the sample variance estimator. It would be helpful to rewrite 'even for very large $M$, $σ^2$ is (with high probability) underestimated by $σ^2$' as 'even for large $M$, the sample variance $σ^2$ may be unstable or the normal asymptotic theory may not apply due to skewness' because the sample variance is a consistent estimator and does not asymptotically underestimate the true variance.

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### 15. Magnitude of Underestimation for GT at theta=15.0

**Status**: [Pending]

**Quote**:
> 0000 samples, the samples from $Q_{\theta}^{\mathrm{GT}}$ underestimate the likelihood by seven orders of magnitude. The IS interpretat

**Feedback**:
The text states that the $Q_{θ}^{“GT”}$ estimate underestimates the likelihood by seven orders of magnitude. Comparing the Table 1 values for $θ = 15.0$, the $Q_{θ}^{“GT”}$ estimate is $2.41 × 10^{-19}$ and the $Q_{θ}^{“SD”}$ (10⁷ samples) estimate is $5.37 × 10^{-12}$. The ratio is approximately $4.5 × 10^{-8}$, which is closer to eight orders of magnitude ($10^{-8}$) than seven ($10^{-7}$). It would be helpful to update the claim to 'nearly eight orders of magnitude' or 'more than seven orders of magnitude' to better reflect the calculated ratio of ≈ $2.2 × 10^7$.

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### 16. Standard Error Scaling Factor Precision

**Status**: [Pending]

**Quote**:
> the importance weights. If they do then the longer run should result in standard errors which are smaller by a factor of $\sqrt{500} \approx 22$. For $\theta = 2.0

**Feedback**:
The text calculates the expected scaling factor for standard errors between the short run (20,000 samples) and the long run (10,000,000 samples) as $√500 ≈ 22$. The exact value of $√500$ is approximately 22.36. Rounding to 22 is acceptable, but given the subsequent discussion relies on the precision of this factor (e.g., comparing observed factors of 21, 17, and <2), using 22.4 or noting the approximation more explicitly would improve clarity. It would be helpful to write '$√500 ≈ 22.4$' to align with the precision of the observed factors reported in the text.

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### 17. Ambiguity in 'Neither Method' Accuracy Claim

**Status**: [Pending]

**Quote**:
> from $Q_{\theta}^{\mathrm{GT}}$ are not. For $\theta = 15.0$, although neither method produces a very accurate estimate of the likelihood using 20000 samples, the sam

**Feedback**:
The text claims that 'neither method produces a very accurate estimate' for $θ = 15.0$ using 20,000 samples. However, the $Q_{θ}^{“SD”}$ (20,000 samples) estimate is $4.74 × 10^{-12}$, which is within approximately 12% of the $Q_{θ}^{“SD”}$ (10⁷ samples) estimate ($5.37 × 10^{-12}$). In the context of likelihood estimation, a 12% error is often considered reasonably accurate, especially compared to the GT result. It would be helpful to clarify that while the SD point estimate is reasonably close, the reported standard error is unreliable.

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### 18. Imprecise Terminology for Standard Error and Standard Deviation

**Status**: [Pending]

**Quote**:
> suggesting that the standard errors in the smaller sample are accurately reflecting the standard deviation of the weights

**Feedback**:
The standard error (SE) of an estimator is defined as the standard deviation of the sampling distribution of the estimator, which equals the standard deviation of the weights ($	ext{SD}_w$) divided by $	ext{N}^{0.5}$. The text implies the SE directly reflects the SD of the weights, which is conceptually imprecise. It would be clearer to state that the SEs are consistent with the theoretical scaling derived from the standard deviation of the weights. It would be helpful to rewrite 'suggesting that the standard errors in the smaller sample are accurately reflecting the standard deviation of the weights' as 'suggesting that the standard errors in the smaller sample are consistent with the theoretical scaling derived from the standard deviation of the weights' to ensure mathematical precision.

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### 19. Ambiguous Citation Regarding Infinite Variance

**Status**: [Pending]

**Quote**:
> hcal{G}}|A_n)$ in this range. (In fact Stephens (1999) has recently shown that for $\theta > 2 \theta_0$ the estimator has infinite variance.)
> 
> ## 5.4. M

**Feedback**:
This citation appears at the end of a paragraph discussing the Kuhner et al. MCMC method, implying the infinite variance result applies to that estimator. However, Stephens (1999) likely refers to the authors' own work on the Importance Sampling (IS) method. If the result applies to the IS method, it should not be used to explain the failure of the Kuhner method without clarification. It is not clear if the infinite variance claim refers to the Kuhner estimator or the IS estimator. It would be helpful to add a clarifying phrase after 'the estimator' to specify which method is being discussed, such as 'the Kuhner estimator' or 'the IS estimator', to avoid confusion.

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### 20. Potential Bias Claim in Importance Sampling Estimation

**Status**: [Pending]

**Quote**:
>  As a result such methods will tend to underestimate the relative likelihood away from $\theta_{0}$, leading to an estimated curve which is artificially peaked about this driving value (which we believe helps to explain the overly peaked curves in Fig. 2(c) for examp

**Feedback**:
The statement suggests a systematic underestimation (bias) of the likelihood when $θ$ is far from $θ_0$. However, importance sampling estimators are theoretically unbiased ($E[rac{p(x)}{q(x)}] = p(x)$). The practical issue is high variance due to skewness, which can lead to unstable estimates that often fall below the true value in finite samples (due to missing rare high-weight samples), but this is a variance issue, not bias. It would be more accurate to attribute the peaked appearance to high variance and poor tail coverage rather than systematic underestimation. It would be helpful to rewrite 'As a result such methods will tend to underestimate the relative likelihood away from $θ_{0}$, leading to an estimated curve which is artificially peaked about this driving value' as 'As a result such methods may produce highly variable estimates that often fall below the true likelihood away from $θ_{0}$ due to poor tail coverage, leading to an estimated curve which appears artificially peaked about this driving value' because importance sampling estimators are unbiased in expectation.

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### 21. Clarification on Markov Structure Terminology

**Status**: [Pending]

**Quote**:
> the simple Markov structure of the underlying processes is lost

**Feedback**:
The coalescent process with varying population size remains a Markov process, specifically a time-inhomogeneous one. The statement that the 'Markov structure... is lost' could be misinterpreted to mean the process is no longer Markovian. It would be helpful to specify that the time-homogeneous Markov structure is lost, which necessitates tracking time in the backward simulation. It would be helpful to rewrite 'the simple Markov structure of the underlying processes is lost' as 'the time-homogeneous Markov structure of the underlying processes is lost' to ensure mathematical precision.

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### 22. Potential Misattribution of Discarding Strategy in Cited Works

**Status**: [Pending]

**Quote**:
> (This is a more sophisticated version of the strategy of discarding trees with too many mutations which was used by Griffiths and Tavaré (1994a) and Nielsen (1997).)

**Feedback**:
Readers familiar with the standard Griffiths–Tavaré importance sampling scheme may note that trees are typically weighted rather than discarded based on mutation counts. It is not clear how the cited papers employed a discarding strategy as described, as the standard method relies on weighting all simulated genealogies. It would be helpful to verify the citation or clarify the specific heuristic being referenced to avoid misattribution. It would be helpful to rewrite the parenthetical claim to accurately reflect the weighting strategy used in the cited works, or remove the reference to discarding if it does not apply.

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