# Inference in molecular population genetics

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

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## Overall Feedback

Here are some overall reactions to the document.

**Unestablished variance finiteness for importance sampling weights**

Readers might note that the paper acknowledges in Section 6.4 that the variance of importance sampling weights is proven finite only for the infinite sites model, leaving unproven the theoretical guarantees for general mutation models such as the stepwise model used in Section 5.4. This gap matters because the central limit theorem-based standard errors reported in Tables 1 and 2 assume finite variance, and Figure 12 demonstrates that these standard errors can severely underestimate true sampling variability when weight distributions are skewed. It would be helpful to establish sufficient conditions for finite variance under general mutation matrices or implement extreme value theory diagnostics and defensive mixture distributions to ensure valid confidence intervals.

**Unquantified approximation error for non-parent-independent mutation models**

Readers might note that Theorem 1 and Proposition 1 establish that the proposed approximation $\hat{\pi}(\cdot|\cdot)$ equals the true conditional probability $\pi(\cdot|\cdot)$ only under parent-independent mutation (PIM) or for sample size $n=1$, yet the method is applied to microsatellite data under the stepwise mutation model which violates this assumption. The paper provides no quantitative bounds on the Kullback-Leibler divergence or other measures of approximation error for such models, meaning the proposal distribution $Q_{\theta}^{\mathrm{SD}}$ may be far from optimal and the claimed efficiency gains rely on empirical demonstration rather than theoretical assurance. It would be helpful to derive approximation error bounds for specific non-PIM mutation models or demonstrate that efficiency gains degrade gracefully as the mutation model deviates from PIM.

**Limited validation of scalability for large-scale and multilocus data**

While the paper demonstrates efficiency gains for small samples ($n \leq 60$) and simple genetic systems, Section 6.5 acknowledges that many real datasets exceed current computational limits. Readers might note that the empirical validation does not include complexity analysis or simulations for larger sample sizes (e.g., $n > 200$), multiple loci with recombination, or genome-scale data where the state space of coalescent histories grows super-exponentially. It would be helpful to provide computational complexity analysis and empirical results for larger samples and multilocus data to establish the practical limits of the method for modern genomic applications.

**Sensitivity to driving value without systematic multi-$\theta$ estimation**

The importance sampling estimator in equation (12) requires a fixed driving value $\theta_0$, and Section 6.1 notes that efficiency degrades when estimating likelihoods at $\theta$ far from $\theta_0$, with potential infinite variance when $\theta > 2\theta_0$. Readers might note that the paper lacks systematic guidance for selecting $\theta_0$ when the true parameter is unknown and does not implement bridge sampling or path sampling methods to combine estimates from multiple driving values reliably, risking artificially peaked likelihood surfaces and underestimated likelihoods. It would be helpful to incorporate adaptive driving value selection or formal bridge sampling into the methodology to ensure accurate likelihood surface estimation across the parameter space.

**Identifiability limitations of the composite parameter $\theta$**

The methodology focuses inference on the composite parameter $\theta = 4N_e\mu$, confounding the effective population size $N_e$ and mutation rate $\mu$ without addressing the fundamental identifiability problem that prevents their separate estimation from single-locus data. Readers might note that this limitation restricts biological interpretability, as external calibration or multilocus information is required to distinguish demographic history from mutational processes. It would be helpful to explicitly discuss this constraint in the interpretation of results and suggest extensions incorporating multiple loci or prior information to identify the constituent parameters.

**Unverified robustness to violations of coalescent assumptions**

The method assumes neutral evolution, panmixia, constant population size, and absence of recombination (Section 2), yet provides no assessment of how violations of these assumptions affect inference efficiency or bias. Readers might note that real populations often experience selection, population structure, and demographic changes that could degrade the proposal distribution's efficiency or introduce bias in $\theta$ estimation, as the derivation is specific to the neutral coalescent. It would be helpful to include sensitivity analyses demonstrating how model misspecification affects estimates or extend the method to handle structured populations and selection as mentioned in Section 6.2.

**Approximation of Conditional Probabilities for Non-Parent-Independent Mutation Models**

It would be helpful to clarify the theoretical justification for applying the proposed importance sampling function to microsatellite data under the stepwise mutation model. Theorem 1 characterizes the optimal proposal distribution using the conditional sampling probabilities $\pi(\beta|A_n)$ defined in equation (14). As noted in Section 4, these probabilities are known exactly only for the special case of parent-independent mutation (PIM), where $P_{\alpha\beta} = P_{\beta}$ for all $\alpha, \beta$ (equation 16). For the stepwise mutation model used in Section 5.4, where mutations increase or decrease allele length by 1 with equal probability, the transition probabilities depend explicitly on the parental type ($P_{i,i+1} = P_{i,i-1} = 1/2$), violating the PIM assumption. 

Consequently, the optimal proposal distribution $Q_{\theta}^{*}$ is intractable for this data type. The authors instead employ the approximation $\hat{\pi}$ (Definition 1), which Proposition 1 shows is exact only under PIM or for sample size $n=1$ with reversibility. While the empirical results in Figure 3 demonstrate practical efficiency, the theoretical optimality guarantees established in Theorem 1 do not apply to the stepwise model. It is not clear how the approximation $\hat{\pi}$ behaves for stepwise mutation with large $\theta$ or boundary effects near the truncation points (0 and 19 repeats). A robustness check comparing the efficiency of the proposed method against simulations under the exact stepwise model (rather than PIM) would help validate the approximation for this genetic system.

**Demographic Model Assumptions for Structured Human Populations**

Readers might note that the application to the NSE microsatellite data (Section 5.4) appears to conflict with the paper's formal demographic assumptions. Section 2 specifies a model with "constant (large) size $N$ chromosomes throughout its history" and implicitly assumes panmixia (random mating) within a single population. However, the NSE dataset combines samples from three distinct human populations (Nigeria, Sardinia, and East Anglia) that are known to exhibit genetic structure and have experienced distinct demographic histories including bottlenecks and expansions.

This mismatch means the scaled mutation rate $\theta = 2N\mu$ (for the haploid Y-chromosome) cannot be interpreted as a single, constant parameter across the sample. The coalescent process for structured data follows a "structured coalescent" with different coalescent rates within and between populations, which violates the exchangeability assumption underlying the proposal distribution in equation (25). As a result, estimates of $\theta$ may be biased or may represent an undefined composite of population sizes and migration rates. It would be helpful to reference the extensions to structured populations (cited in Section 6.2 as Bahlo and Griffiths, 2000) when analyzing such heterogeneous samples, or to restrict analysis to single-population subsamples to maintain consistency with the assumed model.

**Status**: [Pending]

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## Detailed Comments (19)

### 1. Incorrect formula enumerating tree topologies

**Status**: [Pending]

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

**Feedback**:
Readers might note that the expression $n!(n-1)!/2^{n-1}$ enumerates labeled histories (ranked tree topologies) rather than unranked topologies. For $n=4$, this yields 18, whereas the number of unranked topologies is $(2n-3)!! = 15$. It would be helpful to clarify that this formula counts labeled histories, or replace it with $(2n-3)!!$ if unranked topologies are intended.

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### 2. Missing factor in Griffiths–Tavaré recursion coalescence term

**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**:
Readers might note that the coalescence term in equation (34) appears to omit a factor of $n_\alpha/n$. The probability of coalescing two lineages of type $\alpha$ requires choosing 2 from $n_\alpha$ and 2 from $n$, yielding $\binom{n_\alpha}{2}/\binom{n}{2} = n_\alpha(n_\alpha-1)/(n(n-1))$, whereas the equation shows $(n_\alpha-1)/(n-1)$. It would be helpful to verify whether the factor $n_\alpha/n$ was omitted intentionally or should be included as $\frac{n_\alpha(n_\alpha-1)}{n(n-1)}$.

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### 3. Inconsistent mutation rate specification in Appendix

**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**:
Readers might note that Appendix A states a per-locus mutation rate of $\theta/2$, but equation (32) uses a Poisson mean of $\theta t/n$, which corresponds to rate $\theta$ rather than $\theta/2$ when integrated against the exponential time $t$. This inconsistency suggests the rate should be specified as $\theta$ to align with the derivation in equations (31)-(33) and Definition 1.

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### 4. Unqualified optimality claim for proposal distribution

**Status**: [Pending]

**Quote**:
> ative likelihood, appear to make things unnecessarily difficult for themselves (see Stephens (1999)). Since the IS function which these MCMC methods use is most efficient (in fact optimal) at $\theta_{0}$, and tends to become less efficient for $\theta$ away from $\theta_{0}$, the distribution of the importance weights tends to be more skewed for $\theta$

**Feedback**:
Readers might note that the claim the IS function is 'optimal' at $\theta_0$ holds strictly only under parent-independent mutation (PIM), per Proposition 1. For the stepwise mutation model used in Section 5.4, $\hat{\pi}$ is an approximation. It would be helpful to qualify this claim as 'optimal under PIM' or 'approximately optimal' to avoid overstating the theoretical guarantee.

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### 5. Incorrect summation index in importance sampling weights

**Status**: [Pending]

**Quote**:
> Indeed, it follows from standard IS theory that the distribution with atoms of size $w_{i} / \sum_{i} w_{j}$ on histories $\mathcal{H}^{(i)}$ ($i = 1, \ldots, M$) is an approximation to the conditional distribution of $\mathcal{H}$ given $A_{n}$.

**Feedback**:
Readers might note that the normalized weight expression $w_i / \sum_i w_j$ contains inconsistent indices, with $i$ as the summation index but $j$ in the summand. This should read $w_i / \sum_j w_j$ to correctly denote normalization by the sum over all $j$.

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### 6. Typographical error in driving value notation

**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**:
Readers might note that Section 5.4 uses $\pi_0$ as the driving value parameter, whereas the standard notation throughout the paper is $\theta_0$. It would be helpful to correct $\pi_0$ to $\theta_0$ to maintain notational consistency.

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### 7. Notation clash for conditional probability symbol $\pi$

**Status**: [Pending]

**Quote**:
> ation on the type of the MRCA. Using IS as before gives
> 
> $$
> P _ {\theta} \left(A _ {n}, O\right) \approx \frac {1}{M} \sum_ {i = 1} ^ {M} \pi_ {\theta} \left(A _ {n} \mid \mathcal {H} ^ {(i)}\right) P _ {\theta} \left(O \mid \mathcal {H} ^ {(i)}\right) \frac {P _ {\theta} \left(\mathcal {H} ^ {(i)}\right)}{Q _ {\theta} \left(\mathcal {H} ^ {(i)}\right)}. \tag {30}
> $$
> 
> Typically it is reasonable to assume that $P_{\theta}(O|\mathcal{H})$ depends only on the root $r(\mathcal{H})$ of $\mathcal{H}$ and i

**Feedback**:
Readers might note that $\pi_\theta(\cdot|\cdot)$ denotes both the conditional sampling distribution of an $(n+1)$th chromosome (Theorem 1, equation (14)) and the likelihood of data given history (equation (30)). It would be helpful to use distinct notation, such as $P_\theta(A_n|\mathcal{H})$, for the latter to avoid conflating these distinct conditional probabilities.

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### 8. Ambiguous equation reference for conditional distribution

**Status**: [Pending]

**Quote**:
> igh dimensional stochastic processes.
> 
> Useful insights for further improving the transition rates in IS proposal distributions might be gained by examining those of the optimal proposal distribution in small problems in which the latter can be calculated using good estimates of the conditional distribution $\pi$ in equations (15). Since the likelihood in question is the solution of a system of linear equations, it may also be fru

**Feedback**:
Readers might note that the reference to $\pi$ in 'equations (15)' appears incorrect, as $\pi$ is defined in equation (14) while equation (15) defines the optimal backward transition probabilities $q^*$. It would be helpful to correct this reference to equation (14).

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### 9. Ambiguity in magnitude of reported efficiency gains

**Status**: [Pending]

**Quote**:
> Our approach substantially outperforms existing IS algorithms, with efficiency typically improved by several orders of magnitude.

**Feedback**:
Readers might note that the summary claims efficiency improvements of 'several orders of magnitude,' but Section 5.2 reports standard error reductions of approximately 17-22 (roughly 1.3 orders of magnitude for standard error, or 2.6 for variance). It would be helpful to revise the claim to 'two to three orders of magnitude in variance' to align precisely with the empirical results.

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### 10. Ambiguity in stationary distribution uniqueness conditions

**Status**: [Pending]

**Quote**:
> To simplify the description we focus on the case where $P$ has a unique stationary distribution.

**Feedback**:
Readers might note that the text assumes $P$ has a unique stationary distribution without specifying sufficient conditions. For a finite Markov chain, uniqueness requires irreducibility and aperiodicity. It would be helpful to add 'assuming $P$ is irreducible and aperiodic' to ensure the ergodicity conditions are explicit.

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### 11. Notational clarity between conditional likelihood and joint density

**Status**: [Pending]

**Quote**:
> If $\mathcal{T}$ represents missing data for which $\pi_{\theta}(A_n | \mathcal{T})$ is (relatively) easy to calculate, then it is straightforward to use the Metropolis–Hastings algorithm to construct a Markov chain with stationary distribution $P_{\theta}(\mathcal{T} | A_n)$. If $\mathcal{T}^{(1)}, \mathcal{T}^{(2)}, \ldots$ is a sample (after burn-in, and possibly thinning) from a Markov chain with stationary distribution $P_{\theta_0}(\mathcal{T} | A_n)$, then a relative likelihood surface for $\theta$ can be estimated by using

**Feedback**:
Readers might note that Section 3.2 introduces $\pi_\theta(A_n|\mathcal{T})$ as the conditional likelihood but switches to $P_\theta(\mathcal{T}, A_n)$ for the joint density in equation (7) without explicit factorization. It would be helpful to state that $P_\theta(\mathcal{T}, A_n) = P_\theta(\mathcal{T})\pi_\theta(A_n|\mathcal{T})$, noting that the coalescent prior $P_\theta(\mathcal{T})$ must also be tractable for MCMC feasibility.

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### 12. Ambiguity in contrast between MCMC and Griffiths–Tavaré likelihoods

**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**:
Readers might note that Section 3.3 claims MCMC approaches give only relative likelihoods 'in contrast with' the Griffiths–Tavaré scheme, yet Section 3.1 suggests the Griffiths–Tavaré method also produces quantities proportional to the likelihood. It would be helpful to clarify whether the Griffiths–Tavaré scheme estimates absolute likelihoods (including normalizing constants) or relative likelihoods, ensuring the contrast is accurate.

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### 13. Clarification needed on estimator (9) versus estimator (8)

**Status**: [Pending]

**Quote**:
> Thus our IS estimator (9) should have zero variance and give the exact value for the likelihood, which in fact it does.

**Feedback**:
Readers might note that Section 5.1 refers to estimator (9) having zero variance for PIM models, but Section 5 discusses estimator (8) as asymptotically normal. It would be helpful to clarify that estimator (9) refers to the likelihood estimator $\hat{L} = M^{-1}\sum w_i$, distinguishing it from the variance discussion in Section 5, and noting that zero variance applies specifically to the likelihood (not log-likelihood) estimator.

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### 14. Ambiguity in single-chain likelihood estimation method

**Status**: [Pending]

**Quote**:
> Naïve application of this method to estimate the likelihood function $L(\cdot)$ requires simulation from a different Markov chain for each value of $\theta$. Griffiths and Tavaré (1994a) showed how to use realizations of a single Markov chain to estimate the likelihood function.

**Feedback**:
Readers might note that Section 3.1 contrasts the 'naïve' approach (different chains for each $\theta$) with the Griffiths–Tavaré single-chain method without explaining that the latter requires importance sampling weights to reweight realizations via the Radon–Nikodym derivative. This omission obscures the fact that both methods rely on importance sampling, differing only in whether the sampling distribution is held fixed.

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### 15. Invalid parametric bootstrap under infinite variance

**Status**: [Pending]

**Quote**:
> In particular we propose 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 (i.e. the estimate of the likelihood).

**Feedback**:
Readers might note that the proposed parametric bootstrap for GPD-fitted weights (Section 6.4) assumes finite variance, which requires the fitted shape parameter $\xi < 1/2$. If the true distribution has infinite variance ($\xi \geq 1/2$), the bootstrap variance estimator diverges. It would be helpful to add the condition that this procedure is valid only if $\hat{\xi} < 1/2$, and recommend subsampling methods otherwise.

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### 16. Autocorrelation in MCMC samples for density estimation

**Status**: [Pending]

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

**Feedback**:
Readers might note that Section 3.3 suggests smoothing MCMC samples for density estimation without addressing autocorrelation. Standard kernel methods assume independence, and applying them to correlated samples may produce undersmoothed, artificially narrow density estimates. It would be helpful to recommend thinning the chain or using specialized bandwidth selectors for dependent data.

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### 17. Ambiguity in coalescent time scaling for Fluctuate comparison

**Status**: [Pending]

**Quote**:
> Instead of considering the typed ancestry of the sample as missing data, they considered the unknown scaled genealogical tree $\tilde{\mathcal{G}}$, i.e. the genealogical tree $\mathcal{G}$ with branch lengths scaled by the mutation rate $\theta / 2$, so that mutations may be assumed to occur at unit rate along the branches.

**Feedback**:
Readers might note that Section 5.3 describes scaling branch lengths by $\theta/2$ for the Fluctuate comparison, but the appropriate scaling depends on the coalescent time units. If time is measured in $4N_e$ generations (diploid), scaling by $\theta = 4N_e\mu$ achieves unit mutation rate, whereas $\theta/2$ would be correct for haploid models with time in $2N_e$ generations. It would be helpful to specify the time units assumed to verify the scaling factor.

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### 18. Ambiguous scalability claims regarding computational limits

**Status**: [Pending]

**Quote**:
> Although the field is still at an early stage, inference in these contexts is already important in practice, and methods described in the paper are becoming routinely used by biologists for much larger data sets, and in some more complex settings. The imminent completion of many of the current genome projects will be followed by an explosion of data documenting molecular variation in natural populations. Indeed, many real data sets are at or beyond the computational limits of current algorithms.

**Feedback**:
Readers might note that Section 6.5 claims methods are used for 'much larger data sets' but immediately states many datasets are 'at or beyond the computational limits.' This juxtaposition creates ambiguity about whether the methods are currently feasible for large-scale data or are limited to smaller datasets. It would be helpful to clarify the practical boundaries of the method's applicability.

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### 19. Undefined term 'constrained' limits immediate comprehension

**Status**: [Pending]

**Quote**:
> vement in efficiency (typically by several orders of magnitude) and accuracy. We also compare our method with existing MCMC approaches. In problems in which the genetic structure is ‘constrained' (in a sense which will become clearer later) IS seems competitive with, or superior to, these MCMC approaches. The latter seem to have an advantage for less constrained problems. There a

**Feedback**:
Readers might note that Section 1 describes genetic structure as 'constrained' without immediate definition, deferring explanation to later sections. While the parenthetical acknowledges this deferral, the term's meaning—whether referring to limited haplotype diversity, parent-independent mutation, or low mutation rates—remains ambiguous until Section 4. It would be helpful to briefly instantiate the term with an example, such as 'under the infinite sites model or with limited haplotype diversity,' to clarify the scope of the IS versus MCMC comparison.

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