# Inference in molecular population genetics

**Date**: 03/06/2026
**Domain**: statistics/statistical_methods
**Taxonomy**: academic/review_paper
**Filter**: Active comments

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## Overall Feedback

Here are some overall reactions to the document.

**Approximation Error Bounds for General Mutation Matrices**

The proposal distribution relies on an approximation of the conditional sampling probability that is proven exact only for parent-independent mutation or sample size $n=1$. It would be helpful to derive an upper bound on the divergence between the approximate and exact probabilities for general mutation matrices to quantify potential bias in likelihood estimates.

**Finiteness of Importance Weight Variance**

The authors acknowledge an inability to prove the finiteness of weight variance except in the infinite sites model, which raises questions regarding the validity of standard error calculations relying on the Central Limit Theorem. Readers might note that skewed weight distributions can lead to underestimation of uncertainty, and it would be beneficial to establish conditions for finite variance or adopt robust variance estimation techniques.

**Sensitivity of Likelihood Surfaces to Driving Value**

The primary results utilize a single driving value for the importance sampling estimator without demonstrating robustness across a wide range of parameter values. It would be helpful to implement strategies such as bridge sampling or multiple driving values to clarify whether the method's performance is sensitive to the choice of $\theta_0$ in practical applications.

**Generalizability to Complex Demographic and Genetic Scenarios**

The theoretical results assume constant population size and neutrality, while empirical validation excludes recombination despite claims of applicability to modern data. It would be helpful to include simulation results assessing bias when the true model deviates from these assumptions or to quantify the impact of ignoring recombination in the examples.

**Validation of Convergence Diagnostics**

The proposed stopping rule based on fitting a Generalized Pareto distribution to weights is described as inexact and lacks rigorous validation against known likelihood values. Developing a validated diagnostic framework that accounts for the heavy-tailed nature of importance weights would improve the practical utility of the method for users.

**Formal Derivation for Infinite Sites Model Extension**

The extension to the infinite sites model is introduced by analogy rather than through a rigorous derivation from the underlying stochastic process. A formal derivation of the backward transition rates for this context would clarify the theoretical properties and ensure efficiency gains translate correctly from the finite type space.

**Reliance on Finite Variance for IS Standard Errors**

Section 5 states that the IS estimator is asymptotically normal with variance $σ²/M assuming finite variance of the weights. However, Section 6.4 notes that finiteness of the variance "could not prove... except in the special case of the infinite sites model." Standard errors are reported for sequence and microsatellite examples (e.g., Table 1, Fig. 3) where this proof is lacking. It would be helpful to clarify the conditions under which the variance is finite for these general models, or to qualify the interpretation of the reported standard errors given the theoretical uncertainty.

**Demographic Model Assumptions vs. Real Data Characteristics**

The formal model assumes a constant population size and stationarity (Section 2). The NSE dataset used in Section 5.4 consists of real human Y-chromosome data, which likely violates these assumptions due to known population expansions and structure. While extensions to varying population sizes are mentioned (Section 6.2), the NSE results are presented under the constant size assumption. Readers might note that the inferred parameter $θ$ could be confounded with demographic history in this application, and it would be helpful to discuss the robustness of the estimates to such misspecification.

**Status**: [Pending]

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## Detailed Comments (25)

### 1. Notation Collision in Transition Probabilities

**Status**: [Pending]

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

**Feedback**:
Readers might note that the symbol $n$ is defined earlier in the section as the fixed sample size, but Equation (1) redefines it as the variable number of chromosomes in $H_{i-1}$. It would be helpful to use a distinct symbol (e.g., $k$) for the current lineage count to avoid confusion in the denominators.

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### 2. Transcription Error in Proposition 1(c) Derivation

**Status**: [Pending]

**Quote**:
> Definition 1 gives
> 
> $$
> \begin{array}{l}
> \hat {\pi} (\beta | A _ {n}) = \sum_ {\alpha \in E} \sum_ {m = 0} ^ {\infty} \frac {n _ {\alpha}}{n} \left(\frac {n}{n + \theta}\right) ^ {m} \frac {\theta}{n + \theta} (P ^ {m}) _ {\alpha \beta} \\
> = \sum_ {\alpha \in E} \sum_ {m = 0} ^ {\infty} \frac {n _ {\alpha}}{n} (1 - \lambda_ {n}) (\lambda_ {n} P) _ {\alpha \beta} ^ {m} \\
> = \sum_ {\alpha \in E} \frac {n _ {\alpha}}{n} (1 - \lambda_ {n}) (I - \lambda_ {n} P) _ {\alpha \beta} ^ {- 1},
> \end{array}
> $$

**Feedback**:
The proof of Proposition 1(c) appears to swap the terms in the geometric distribution compared to Definition 1. It would be helpful to correct the first line of the derivation to match Equation (17), specifically changing $(\frac{n}{n+\theta})^m \frac{\theta}{n+\theta}$ to $(\frac{\theta}{n+\theta})^m \frac{n}{n+\theta}$.

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### 3. Discrepancy in Griffiths–Tavaré Likelihood Estimates

**Status**: [Pending]

**Quote**:
> In our own implementation of the Griffiths–Tavaré proposal distribution, 1 million samples took 72 h and gave an estimate of the likelihood (dominated by a single large importance weight) of $4.93 \times 10^{-13}$.

**Feedback**:
The text reports an estimate of $4.93 \times 10^{-13}$ for the Griffiths–Tavaré method, while Table 1 lists $2.41 \times 10^{-19}$ for the same parameter. It would be helpful to clarify why the authors' re-implementation differs significantly from the original published values in Table 1.

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### 4. Inconsistent Parameter Symbol in Figure Captions

**Status**: [Pending]

**Quote**:
> using micsat and $Q_{\eta_0 = 10.0}^{\mathrm{SD}}$

**Feedback**:
The captions for Figures 4 and 5 use the symbol $\eta_0$ for the driving value, whereas the main text consistently uses $\theta_0$. It would be helpful to standardize the notation to $\theta_0$ throughout to align with the defined mutation rate parameter.

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### 5. Terminology Error Regarding 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**:
The text states there are $S + 1$ rooted gene trees, but in the infinite sites model, there are $S + 1$ possible roots (ancestral sequences), not trees. It would be helpful to correct this terminology to 'roots' to align with standard definitions.

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### 6. Notation Inconsistency for Proposal Distribution

**Status**: [Pending]

**Quote**:
> This procedure defines an IS function $Q^{\mathrm{SD}}$ which we note is independent of $\theta$

**Feedback**:
The text states the IS function $Q^{\mathrm{SD}}$ is independent of $\theta$, but Equation (30) uses the notation $Q_{\theta}$. It would be helpful to update Equation (30) to $Q^{\mathrm{SD}}$ or $Q$ to reflect this independence.

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### 7. Ambiguous Summation Notation in IS Weights

**Status**: [Pending]

**Quote**:
> distribution with atoms of size $w_i/\Sigma_i w_j$

**Feedback**:
The notation $\Sigma_i w_j$ uses the summation index $i$ which conflicts with the free index $i$ for the history $\mathcal{H}^{(i)}$. It would be helpful to rewrite this as $\sum_{j=1}^M w_j$ to avoid notational inconsistency.

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### 8. Missing Normalization for Ascertainment Bias

**Status**: [Pending]

**Quote**:
> For the IS schemes that we have considered, assuming the infinite sites mutation model, the ascertainment effect can be accommodated by labelling every lineage which leads to any chromosome in the panel as a panel lineage and adapting both $P_{\theta}(\mathcal{H})$ and $Q_{\theta}(\mathcal{H})$ so that mutations can occur only on panel lineages.

**Feedback**:
The description of accommodating ascertainment bias restricts the process but omits normalization by the probability of the ascertainment event $P(S)$. It would be helpful to add 'and normalizing by the probability of the ascertainment event $P(S)$' to ensure the likelihood is valid.

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### 9. Mutation Rate Definition Mismatch in Appendix A

**Status**: [Pending]

**Quote**:
> We assume for simplicity that each locus mutates independently at rate $\theta / 2$ (so the total rate is $l\theta / 2$), according to a $k \times k$ transition matrix $P$.

**Feedback**:
Appendix A states the mutation rate is $\theta/2$ per locus, but Equation (32) implies a rate of $\theta/n$ via the Poisson mean. It would be helpful to align the text description with the equation, likely changing the rate to $\theta/n$.

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### 10. Invalid Likelihood Approximation in Discussion

**Status**: [Pending]

**Quote**:
> Thus an approximation to $L(\theta)$ that utilizes the summation over (distinct) $\mathcal{H}$s visited by the Markov chain is available; such an approximation would be valid for all $\theta$, and thus gives a complete summary of $L(\theta)$.

**Feedback**:
The discussion claims that summing over visited $\mathcal{H}$s yields a valid approximation for all $\theta$, but samples from the joint posterior require importance weighting for likelihood estimation. It would be helpful to clarify that importance weighting or posterior density estimation is necessary for validity.

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### 11. Ambiguity in GPD-Based Mean Estimation

**Status**: [Pending]

**Quote**:
> fitting a generalized Pareto distribution (see Davison and Smith (1990) for example) to the weights above some threshold, and using a parametric bootstrap to estimate confidence intervals for the mean of the weights

**Feedback**:
The proposal to fit a GPD to the tail of the weight distribution does not specify how the bulk of the distribution is treated for the mean estimate. It would be helpful to clarify whether the bootstrap samples the full distribution or only the tail, and how the mean is aggregated.

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### 12. Algorithm 1 Simulation Step Redundancy

**Status**: [Pending]

**Quote**:
> if there are fewer than $n + 1$ lines in the ancestry return to step 2. Otherwise go back to the last time at which there were $n$ lines in the ancestry and stop.

**Feedback**:
Algorithm 1 simulates to $n+1$ lines and then reverts to $n$, which differs from standard coalescent sampling that terminates at $n$. It would be helpful to justify this additional step or align the algorithm with standard procedures to avoid confusion.

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### 13. Product Limits in Griffiths–Tavaré Equation

**Status**: [Pending]

**Quote**:
> n the 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 par

**Feedback**:
The product in Equation (6) runs from $j=0$ to $\tau$, where $\tau$ is the stopping time. In standard formulations, the likelihood is typically expressed as a product over the $\tau$ transitions rather than the $\tau+1$ states. It would be helpful to clarify whether $F$ represents state weights or transition weights and adjust the product limits accordingly.

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### 14. Support Condition for Proposal Distribution

**Status**: [Pending]

**Quote**:
> )$ is any distribution on ancestries whose support includes $\{\mathcal{H} \colon \pi_{\theta}(A_n|\mathcal{H}) > 0\}$

**Feedback**:
The condition for the proposal distribution $Q_{\theta}$ to yield an unbiased estimator is that its support must cover the support of the integrand $\pi_{\theta}(A_n|\mathcal{H}) P_{\theta}(\mathcal{H})$. It would be helpful to refine this to the joint support to be mathematically precise.

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### 15. Weight Variance Formula Applicability

**Status**: [Pending]

**Quote**:
> A natural measure of the variability is then given by the standard error $\hat{\sigma} / \sqrt{M}$, where $\hat{\sigma}^2$ is the sample variance of the $M$ weights.

**Feedback**:
Readers might note that the variance formula $\sigma^2 / M$ applies specifically to the sample mean of unnormalized weights. It would be helpful to clarify whether the weights in estimator (8) are normalized or unnormalized to ensure the variance formula is applicable.

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### 16. Inconsistent Acronym Capitalization

**Status**: [Pending]

**Quote**:
> designing a single iS function $Q_{\theta_0}$

**Feedback**:
The acronym for Importance Sampling is consistently written as 'IS' in the first paragraph, but appears as 'iS' in the fifth and sixth paragraphs. It would be helpful to rewrite 'iS' as 'IS' throughout the section to maintain consistent notation.

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### 17. Inconsistent Missing Data Notation

**Status**: [Pending]

**Quote**:
> clude in the missing data, with the usual trade-offs (typically, the lower the dimension of the missing data $\mathcal{T}$, the harder it is to calculate $\pi_{\theta}(A_n|\mathcal{T})$). we no

**Feedback**:
Readers might note that the section defines the missing data variable as $\mathcal{H}$ in Equation (4), but later refers to 'missing data $\mathcal{T}$' when discussing dimensionality trade-offs. It would be helpful to rewrite this to use $\mathcal{H}$ to maintain consistent notation throughout the section.

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### 18. LaTeX Product Symbol Typo

**Status**: [Pending]

**Quote**:
> averaging the realized values of $\Pi_{j=0}^{\tau} F(B_j)$ across the simulated realizations of the chain.

**Feedback**:
The text description uses the LaTeX command `\Pi` to denote the product, whereas Equation (6) correctly uses `\prod`. In LaTeX, `\Pi` renders as a variable symbol rather than an operator. It would be helpful to ensure the text description matches the mathematical operator used in the equation.

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### 19. MCMC Autocorrelation in Density Estimation

**Status**: [Pending]

**Quote**:
> and dividing by the prior density. The posterior density is most easily estimated by smoothing a sample $\theta^{(1)}, \ldots, \theta^{(M)}$ from the Markov chain, although potentially mo

**Feedback**:
Standard kernel density estimation assumes independent samples, but MCMC samples are typically autocorrelated. It would be helpful to acknowledge this limitation and suggest adjusting for autocorrelation to ensure accurate posterior density estimation.

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### 20. Classification of Griffiths–Tavaré Scheme

**Status**: [Pending]

**Quote**:
> In contrast with the Griffiths–Tavaré scheme described above, and the IS methods described below, these MCMC approaches give only relative, rather than absolute, likelihood surfaces

**Feedback**:
The Griffiths–Tavaré scheme is itself an importance sampling method, so grouping it separately from 'IS methods' may be confusing to readers familiar with the literature. It would be helpful to clarify that GT is an IS method or rephrase to avoid implying a categorical distinction.

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### 21. Cross-Reference to Equation (7)

**Status**: [Pending]

**Quote**:
> with IS (via equation (7)) to estimate the likelihood surface

**Feedback**:
The summary paragraph references equation (7) to describe a framework involving MCMC and IS. However, equation (7) is not provided in the current section text. It would be helpful to ensure equation (7) is defined in the preceding text or clarify the reference here.

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### 22. Clarification on Independent Sampling Mechanism

**Status**: [Pending]

**Quote**:
> for PIM our IS is actually independent sampling from the full conditional distribution of the history of the sample

**Feedback**:
It is not clear how the independence of samples is guaranteed without specifying the sampling mechanism. It would be helpful to clarify whether the algorithm uses a direct sampler or MCMC steps to ensure the 'independent sampling' claim is substantiated.

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### 23. Standard Error Terminology Precision

**Status**: [Pending]

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

**Feedback**:
It would be helpful to clarify that the standard error reflects the standard deviation of the estimator, not the weights directly. The standard error is calculated as the standard deviation of the weights divided by the square root of the sample size.

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### 24. Qualification of Efficiency Claim

**Status**: [Pending]

**Quote**:
> efficiency typically improved by several orders of magnitude

**Feedback**:
The summary asserts a specific quantitative improvement in efficiency without qualifying the conditions under which this holds. Given that the text later acknowledges the method compares 'less favourably' with MCMC in some problems, it would be helpful to clarify the scope of this improvement.

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### 25. Attribution of Importance Sampling Method

**Status**: [Pending]

**Quote**:
> The Griffiths–Tavaré approach has recently been shown (Felsenstein et al. 1999) to be a version of importance sampling (IS).

**Feedback**:
Readers might note that the 1994 paper explicitly describes the method as importance sampling, making the 1999 citation potentially confusing regarding the historical development. It would be helpful to clarify whether the 1999 reference refers to a specific characterization or if the citation should be updated.

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