Simulation-consistent Estimation of the Marginal Likelihood for Block Models

Adrian E. Raftery; Guilhem Fouetillou; Marie Perrot-Dock\`es; Martin Metodiev; Pierre Latouche

REVIEW 3 major objections 5 minor 17 references

Simulation-consistent Estimation of the Marginal Likelihood for Block Models

T0 review · 3 major / 5 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read This paper introduces the truncated harmonic mean estimator (THAMES), a simulation-consistent estimator of the marginal likelihood for stochastic block models that avoids evaluating the likelihood and handles label switching.

desk verdict A clever and potentially useful RIS estimator for SBM marginal likelihoods, but the real-data analysis violates its own consistency guarantee by selecting the best of 10 chains, and the asymptotics ignore data-dependent tuning. read the letter →

arxiv 2607.23998 v1 pith:6C4X3EGF submitted 2026-07-27 stat.ME

Martin Metodiev , Marie Perrot-Dockès , Guilhem Fouetillou , Pierre Latouche , Adrian E. Raftery This is my paper · ORCID

classification stat.ME MSC 62F1562H3062M05

keywords marginallikelihoodstochasticblockmodelselectionharmonicmeanestimatorreciprocalimportancesamplinglabelswitchingMCMCBayesianclustering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces THAMES, the first estimator of the marginal likelihood for stochastic block models that is simulation-consistent: as the MCMC sample size grows, it converges to the true marginal likelihood even when the network size is fixed. This matters because choosing the number of clusters in a network is a Bayesian model selection problem, and existing variational or asymptotic approximations lack finite-sample consistency guarantees. The estimator uses only the MCMC draws of the cluster allocation vectors, exploiting the analytic collapsed prior and collapsed likelihood, so it never evaluates the intractable SBM likelihood. It is asymptotically normal and symmetric under label permutations, and a computational shortcut keeps the cost manageable even when the number of clusters is large. On a real COP28 Twitter/X network, THAMES selects 12 clusters and exposes a sharp core-periphery structure.

What carries the argument

The load-bearing identity is the reciprocal importance sampling representation: for any density h on allocation vectors, the average of h(C^(t))/(p(Y|G,C^(t))p(C^(t)|G)) over posterior draws converges to 1/Z(G). THAMES chooses h as a symmetrized uniform density on a truncation set B = E ∩ H_alpha, where E is a Cartesian product of node-level candidate sets from the first half of the MCMC output and H_alpha is a highest-posterior-density region; this choice makes the estimator label-switching invariant and approximately variance-optimal. The analytic forms of the collapsed prior and collapsed likelihood supply the unnormalized posterior needed in the denominator. A lexicographic ordering of t

What would settle it

Simulate a network with n=10 and G=2 (where Z(G) can be computed exactly by summing all 1024 configurations), run Gibbs sampling with the same chain-selection protocol used on the COP28 data, and compute THAMES on the selected chain as T grows. If the estimate does not converge to the true Z(G), or if it systematically differs from THAMES computed on the full unselected sample, the chain-selection step is biasing the estimator.

Watch

Extended reading notes

Core claim

The central claim is that the marginal likelihood Z(G) of a stochastic block model can be estimated consistently from a Gibbs sample of allocation vectors alone. The estimator is a reciprocal importance sampling estimator whose proposal h is a mixture of uniform distributions over the label-permuted copies of a truncation set A; the truncation set is chosen as the intersection of a Cartesian product of per-node candidate allocations (from the first half of the sample) with a highest-posterior-density region, which the paper shows minimizes variance among symmetric sets under mild conditions. Because the collapsed distributions p(C|G) and p(Y|G,C) are known analytically, each term is computab

Load-bearing premise

The consistency theorem assumes the MCMC allocation draws follow the posterior p(C|G,Y); in the real-data analysis the authors run ten Gibbs chains and keep only the one with the highest MAP value, which can bias the retained sample away from the posterior and break the guarantee.

Editorial extensions

If this is right

Bayesian model selection for the number of clusters in a network becomes simulation-consistent even for small networks, with confidence intervals available from the asymptotic normality.
Any block model with analytically collapsed allocation distributions — including the latent block model — can use the same estimator, extending consistent evidence computation to co-clustering.
The COP28 analysis indicates that marginal-likelihood-based cluster choice can disagree sharply with criteria like ICL, yielding a less fragmented interpretation of the network's core-periphery structure.
The computational shortcut makes the estimator practical for networks with tens of thousands of nodes and cluster counts well beyond 15, where permutation-averaged alternatives are infeasible.
Because the estimator uses only the allocation vectors, it applies even when the likelihood itself is intractable, as long as Gibbs sampling is possible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to compare THAMES computed on the single highest-MAP chain versus on all chains pooled: if the chain-selection protocol biases the posterior sample, the two estimates should drift apart as T grows, and the pooled version should be closer to the truth in settings with known Z(G).
The variance-optimality of the HPD truncation set presumes no posterior ties; in datasets with near-tied allocation probabilities, a randomized tie-breaking or a smoothed truncation set might change the variance ranking, an effect the paper does not explore.
The empty-cluster recursion suggests a direct route to estimating the marginal likelihood for models with random G by chaining THAMES estimates across G, which would make trans-dimensional model selection consistent as well.
The core-periphery pattern found in the COP28 data implies a diagnostic: when a network's posterior mass is concentrated on a few hub-like accounts, marginal-likelihood-based cluster counts may be highly sensitive to the inclusion or exclusion of those hubs; the method's behavior on such sparse, hub-dominated networks deserves specific simulation study.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Good to talk. The short version: the THAMES estimator is a genuine and potentially useful contribution, and the consistency proof is standard and correct for a fixed truncation set. But the real-data section selects the best of 10 Gibbs chains by MAP, which breaks the posterior-sample assumption; and the asymptotics ignore the data-dependent choice of alpha. Both are fixable, but they need attention.

What is actually new: the sample-splitting truncation set, the symmetrization over label permutations via a lexicographic ordering that avoids the G! explosion, and the empty-cluster recursion via Nobile's formula. These are real ideas, and the computational trick is neat. The paper is honest about the strong assumptions behind the optimality of the HPD region (Remark 1), and the proofs in the supplement are reasonably clear.

The soft spots, in proportion: the most serious is in Supplement C.1. Running 10 independent samplers and keeping the chain with the highest MAP biases the retained draws toward high-likelihood modes. Theorem 1 requires draws from the posterior; the COP28 estimates are therefore not covered by the consistency guarantee. This is not a flaw in the estimator, but it means the headline application results are not backed by the theory. Second, the asymptotic normality claim holds only for fixed alpha; the variance-minimization of alpha on the same second-half sample makes the estimator data-dependent in a way that the proof does not address. This is likely fixable with a separate tuning set or a uniformity argument, but as written it's a gap. Third, the simulations are thin: n=10, G=2 only, with no evidence at larger scales where the computational trick matters. Fourth, the novelty claim is slightly overblown—McDaid et al. (2012) already did something similar for a collapsed SBM with random G. The specific THAMES variant is new, but the "first simulation-consistent" phrasing needs a qualifier. Finally, the GitHub link is missing, which hurts reproducibility.

Who is this for: anyone working on Bayesian model selection for block models, especially in network science. The method is worth knowing even if the current application is not fully valid. I would send this to peer review—the core is sound and the issues are addressable—but the authors need to deal with the chain selection before the real-data results can be trusted.

Referee Report

3 major / 5 minor

Summary. The paper proposes THAMES, a truncated harmonic mean estimator for the marginal likelihood of stochastic block models. The estimator is defined as a symmetrized reciprocal importance sampling average over the second half of an MCMC sample, with a truncation set built from the first half using a variational approximation. The authors claim that THAMES is simulation-consistent and asymptotically normal under standard MCMC conditions, invariant to label switching, computationally efficient for large numbers of clusters, and applicable to large networks. The theoretical argument is based on the standard RIS identity (Theorem 1); the truncation set is motivated by an HPD optimality result (Theorem 2); an efficient summation over permutations is given (Theorem 3); and an empty-cluster correction is adapted from Nobile's formula (Theorem 4). The method is tested on n=10, G=2 simulations where the true marginal likelihood is available, and then applied to a COP28 X (Twitter) dataset, yielding G=12 as the preferred number of clusters.

Significance. If the claims hold, this would be a useful contribution: a marginal likelihood estimator for SBMs that does not evaluate the intractable likelihood and that is consistent as the number of MCMC simulations grows, not as the network size grows. The paper also provides a practical device for label-switching invariance and a computational shortcut that avoids the G! factorial sum. The explicit proofs in the supplement, the availability of an R package, and the extension to latent block models are strengths. However, the simulation evidence is limited to a very small setting (n=10, G=2), and the real-data application uses a chain-selection protocol that is not covered by the stated theory. The claimed asymptotic normality also lacks a theorem covering the data-dependent choice of the truncation parameter. These gaps do not invalidate the core RIS idea, but they need to be addressed before the paper's stronger claims can be accepted.

major comments (3)

[Supplement C.1, Table 2] The real-data pipeline runs 10 independent Gibbs samplers per G and retains only the chain with the highest MAP value. Theorem 1 requires C^(1),...,C^(T) to be an ergodic Markov chain with limiting distribution p(C|G,Y). A chain selected by maximizing the unnormalized posterior is not marginally distributed as p(C|G,Y). Because the RIS weight in Eq. (4) is the reciprocal of p(C|G)p(Y|G,C), the selection is informative about the weights: the retained chain is tilted toward large values of this product, so the average of its reciprocals is biased downward as an estimate of 1/Z(G), and log Z(G) is biased upward. The bias can vary with G and is not covered by the consistency guarantee. The COP28 estimates and the choice G=12 therefore rest on an unaddressed violation of the theorem. A single converged chain or a proper combination of multiple chains is needed.
[Section 3.1, Supplement B.1] The truncation parameter alpha is chosen by minimizing the estimated variance of THAMES computed on the same second-half sample used for the point estimate. Theorem 1 and its proof in Supplement A.1 apply to a fixed importance function h, equivalently a fixed truncation set A. When A is selected on the data used for the estimate, the estimator is not a standard RIS estimator, and no theorem is provided for the resulting procedure. The remark in Section 3.1 that 'for a fixed value of alpha' consistency holds does not cover the data-dependent choice. The abstract's asymptotic-normality claim therefore lacks support. The authors should either prove a uniform-in-alpha result for the grid selection, or choose alpha on an independent pilot sample and state the additional conditions needed for the CLT.
[Section 5, Table 2] The paper claims that THAMES allows approximate confidence intervals, but Table 2 reports rounded log-marginal-likelihood estimates with no standard errors. The difference between G=12 and G=13 is only 7 on the log scale. Without Monte Carlo uncertainty quantification, it is not possible to judge whether this difference is meaningful, especially since the estimates are not covered by Theorem 1 because of the chain-selection issue in Supplement C.1. The model-selection conclusion and the subsequent sociological interpretation depend on this table. At minimum, the authors should report standard errors or intervals for all G and discuss the sensitivity of the argmax to Monte Carlo error.

minor comments (5)

[Section 3.3 / Supplement A.6] The empty-cluster correction via Theorem 4 requires p0(G)>0. If the MCMC sample contains no empty clusters, the empirical estimator of p0(G) is 0 and Eq. (12) is undefined. The paper should clarify the procedure in that case, even if the correction is only applied when an empty cluster is observed.
[Section 3.2, Remark 2] The cross-reference 'Equation (25)' in Remark 2 appears to refer to Equation (11) in the main text; similarly, 'Equation (26)' in Section 3.3 should be Equation (12). Please correct the equation numbering in the main text.
[Section 4] The simulation study is limited to n=10, G=2, where the true marginal likelihood has only 1024 terms. This demonstrates the estimator on a toy setting but does not validate the computational or statistical behavior for larger n and G, which is where the method is claimed to be useful. A larger simulation, even without an analytic gold standard, would strengthen the empirical case.
[Supplement C.2] There are several typos: 'connecction' should be 'connection'; in Supplement C.3, 'alocation' should be 'allocation'; 'the Latent dirichlet alocation (LDA) model' is redundant. These do not affect content but should be cleaned up.
[Section 3.2] The statement that the size of Omega 'is usually equal to one in our experience' is informal. Since this is the basis for the claimed efficiency for arbitrarily large G, the authors should either provide a theoretical bound on |Omega| under explicit posterior-concentration conditions or at least report the distribution of |Omega| in the simulations and the COP28 application.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: THAMES is a standard RIS estimator with sample-splitting; consistency follows from Tierney's external theorem, not from the fitted truncation set.

full rationale

The paper's derivation is not circular. THAMES is explicitly defined as a reciprocal importance sampling (RIS) estimator with h_THAMES built from a truncation set A (Section 3, Supplement A.1). Its consistency and asymptotic normality are inherited from Theorem 1, which is Tierney's Markov-chain ergodic theorem, external and machine-independent. The truncation set is estimated from the first half of the MCMC sample and the RIS average is computed on the second half, so the estimator does not evaluate its own fitted quantity: the target Z(G) is fixed by Equation (3), not redefined by the data. The choice of α by variance minimization is a tuning step, not a prediction, and does not make the estimator equal to its inputs. Theorems 2 and 3 are proved in the supplement rather than imported as black boxes; Theorem 4 cites Nobile, an external source. The self-citations (Metodiev et al. 2025a,b) provide computational ordering tricks and a variance estimator, but the central consistency result does not load-bear on them, and the SBM-specific proofs are given. The COP28 best-of-10 MAP chain selection noted in Supplement C.1 is a potential violation of Theorem 1's assumption that draws come from the posterior, and is a correctness/validity concern, but it is not circularity: the estimator's target is still the marginal likelihood, not a re-expression of the selected chain. The simulation studies compare against exactly computed Z(G), providing a benchmark independent of the estimator's own construction. No step in the paper reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard RIS theory and a set of conditions for optimality and efficient computation that the authors explicitly flag as rarely holding exactly. The two data-dependent tuning parameters (α and r̂) are real free inputs that affect the finite-sample behavior and are not accounted for in the asymptotic theory. The empty-cluster recursion adds a chain dependence between estimates of Z(G). No new physical or theoretical entities are postulated.

free parameters (3)

α (HPD truncation level) = not reported; chosen on a grid {0,0.1,...,1} by minimizing estimated variance
The truncation set B = E ∩ H_α depends on α, and α is chosen empirically to minimize the sample variance of the estimator on the second half of the MCMC sample (Supplement B.1). This is a data-dependent tuning parameter.
r̂ (truncation radius / number of candidate allocations per node) = not reported; chosen by minimizing Eq. (23) under a constraint
The set E_{Ĉ,r̂} is defined by r̂ candidate allocations per node taken from the first half of the sample. r̂ is selected by minimizing an analytic variance formula derived under variational assumptions (Supplement B.2), under the condition that the expected number of non-empty terms is at least 20/T.
Grid step for α and the 20/T threshold = grid step 0.1; threshold 20/T
Hand-chosen algorithmic constants that affect the tuning of the estimator. They are not parameters of the statistical model, but they influence the variance and stability of the reported estimates.

assumptions (5)

domain assumption The Gibbs sampler output is an ergodic Markov chain with stationary distribution p(C|G,Y), and the ECR relabelling produces a non-switched sample from the same distribution.
Theorem 1 requires an ergodic chain with the posterior as limiting distribution. The ECR relabelling is assumed to approximately undo label switching without changing the target distribution. In the COP28 analysis, selecting the best of 10 chains adds a further layer of assumption.
domain assumption No two allocation vector matrices have the same posterior probability up to permutations (the 'no ties' condition).
Theorem 2 (HPD optimality) requires this condition. The authors explicitly concede it is not guaranteed and conjecture it holds for sufficiently large/complex datasets. It is not needed for consistency, but it is needed for the reported variance-optimality property.
ad hoc to paper The variational approximation exactly equals the posterior, and the posterior is highly concentrated around one label set (conditions (20) and (21)).
Proposition 1 and the variance formula (Eq. 23) used to select r̂ rely on these assumptions. The authors state these conditions rarely hold exactly and may only hold approximately as n→∞.
domain assumption There are no empty clusters in the second half of the MCMC sample when using the efficient computation of Theorem 3.
Theorem 3 requires no empty clusters. Empty clusters are handled by Theorem 4 (Nobile's formula), but the recursive computation introduces a chain of estimates.
standard math Nobile's formula (Theorem 4) relates Z(G) to Z(G-1) via the posterior probability of an empty cluster, p0(G).
The proof is given in Supplement A.6. It is a known identity for mixture models, extended to the SBM with the Dirichlet/Beta prior setup.

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Cite this review

Pith. "Pith review of Simulation-consistent Estimation of the Marginal Likelihood for Block Models." pith.science (2026). https://pith.science/paper/6C4X3EGF

@misc{pith2026260723998,
  author       = {Pith},
  title        = {Pith review of: Simulation-consistent Estimation of the Marginal Likelihood for Block Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6C4X3EGF}},
  note         = {Machine review of arXiv:2607.23998}
}

read the original abstract

We propose a methodology for computing marginal likelihoods for block models. The proposed estimator computes the marginal likelihood from Markov chain Monte Carlo (MCMC) samples and is simulation-consistent, even when the size of the dataset is fixed. Moreover, it is asymptotically normal, of finite variance, invariant to label switching and can be computed efficiently, even for models with an arbitrarily large number of components. We evaluate the method through simulation studies in settings where the true marginal likelihood is available analytically. Finally, we apply the approach to a social network dataset based on the 2023 United Nations Climate Change Conference (COP28) and discuss the resulting insights.

Figures

Figures reproduced from arXiv: 2607.23998 by the authors.

**Figure 1.** The error log(Z)−log(Zˆ THAMES) of the logarithm of the THAMES for 20 different datasets for each value of λ and for different MCMC sample sizes T, as well as different scenarios (communities, symmetric disassortative mixing, asymmetric disassortative mixing) In this section and the next, we choose the hyperparameters a 0 g,k = b 0 g,k = n 0 1 = · · · = n 0 G = 1, since it has been shown that the posterior distribut… view at source ↗

**Figure 2.** The error log(Z)−log(Zˆ) of the logarithm of the different estimators (the THAMES, the RISVB estimator, the ChibPartition estimator, the harmonic mean estimator and the ILvb estimator as well as its correction, ILvbc) for 20 different datasets as well as different scenarios (communities, symmetric disassortative mixing, asymmetric disassortative mixing) marginal likelihood log(Z) − log(Zˆ THAMES) is shown in [PITH_… view at source ↗

**Figure 3.** The error log(Z) − log(Zˆ) of the logarithm of the THAMES, the RISVB estimator, the ChibPartition estimator, the harmonic mean estimator and the ILvb estimator as well as its correction, ILvbc, for 20 different MCMC simulations on the same dataset; the values of ILvb and ILvbc do not change since they are not calculated on a MCMC sample marginal likelihood well when λ is large, presumably because variational assumpt… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: A graphical representation of the COP28 dataset with 12 clusters, [PITH_FULL_IMAGE:figures/full_fig_p030_4.png]

**Figure 5.** Figure 5: the metagraph of the COP28 dataset for G = 12 clusters, the number that maximises the marginal likelihood estimates of the THAMES; Clusters are represented by nodes. Large edges indicate large connection probabilities between clusters, while large nodes indicate a larg…

**Figure 6.** Figure 6: A heatmap of the probabilities of the 40 different topics (on the y [PITH_FULL_IMAGE:figures/full_fig_p074_6.png]

**Figure 7.** Figure 7: the metagraph of the COP28 dataset for G = 37 clusters, the number obtained by maximising the ICL via the greed algorithm; Clusters are represented by nodes. Large edges indicate large connection probabilities between clusters, while large nodes indicate a large relati…

**Figure 8.** Figure 8: The specific dataset used in Section 4.3 [PITH_FULL_IMAGE:figures/full_fig_p087_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 4 linked inside Pith

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Abbe, E. (2018). Community detection and stochastic block models: recent developments.Journal of Machine Learning Research 18(177), 1–86. Agrawal, S. (2013). Mini-batch-k-means. R package.https://github.com/siddharth-agrawal/ Mini-Batch-K-Means. Aicher, C., A. Z. Jacobs, and A. Clauset (2015, June). Learning Latent Block Structure in Weighted Networks.Jou...

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This probability is much smaller than the usual probabilities obtained when simulating datasets of such a small size, which were often larger than 50 percent, in our experience. Supplement D: defining the THAMES for the latent block model The latent block model (LBM) has a very similar structure to the SBM, with the difference that the size of the rowsnma...

2010
[3]

They are quite variable, with cluster 5, a cluster of size 6222, being the largest cluster, and cluster 11, a cluster of size 5, being the smallest cluster. ˆCM APcan be used to obtain estimates of the SBM parametersµand τby maximising thecomplete-data loglikelihood log(p(Y, ˆCM AP|G, τ, µ)) = nX i=1 GX g=1 ˆCM AP i,g log(τg) + nX i=1 GX g=1 X j̸=i GX k=1...

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Brault, G

Keribin, C., V. Brault, G. Celeux, and G. Govaert (2015, November). Estimation and selection for the latent block model on categorical data.Statistics and Computing 25(6), 1201–1216. Latouche, P., E. Birmel´ e, and C. Ambroise (2014). Model selection in overlapping stochastic block models.Electronic Journal of Statistics 8(1), 762 –

2015
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, C(T) is an ergodic Markov chain with limiting distribution p(C|G, Y), the posterior distribution ofCgivenYandG, then ˆZ−1 RIS(G) almost surely converges toZ −1(G)

IfC (1), . . . , C(T) is an ergodic Markov chain with limiting distribution p(C|G, Y), the posterior distribution ofCgivenYandG, then ˆZ−1 RIS(G) almost surely converges toZ −1(G). IfC (1), . . . , C(T) is additionally ergodic of degree 2, then √ T( ˆZ−1 RIS(G)−Z −1(G))converges in distribution to a centred normal distribution. ProofFollows directly from ...

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The labels of the nodes show the username of the user with the highest number of followers (taken as the maximum of this number over all timepoints on which the user sent a post within this dataset). ”Al Gore” represents cluster 11, whose central function as the core of the dataset is quite clear, as all other clusters are quoting or reposting its users w...

2023
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72 Metodiev et al. A topic-specific summary was obtained as follows: first, an estimate ˆβ is computed via the topicmodels package, where each entry ˆβ˜i,˜j is a matrix that gives the probability that a word with index ˜iappears given the topic with index ˜j(Gr¨ un and Hornik, 2011). Then, the entropy ˆβ˜i,˜j X i′ ˆβi′,˜j log( ˆβi′,˜j)/log( ˜K) +β ˜i,˜j i...

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This may well be explained by topic 5, since this topic is associated with a specific movement, the #SaveSoil movement

This is particularly interesting because cluster 1 is the closest to the sociological definition of a community, with a fairly high internal connection probability. This may well be explained by topic 5, since this topic is associated with a specific movement, the #SaveSoil movement. This movement is also detected when using a different model choice crite...

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Latouche, P., E. Birmel´ e, and C. Ambroise (2012). Variational Bayesian inference and complexity control for stochastic block models.Statistical Modelling 12(1), 93–115. Publisher: SAGE Simulation-consistent Estimation of the Marginal Likelihood for Block Models37 Publication...

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We implemented this algorithm (using the ”kmeans” function from the programming language 64 Metodiev et al

is defined as ILvb = ln{ Γ(PG g=1 n0 g) QG g=1 Γ(ng) Γ(PG g=1 ng) QG g=1 Γ(n0g) } + GX g≤k ln{ Γ(a0 q,lb0 q,l)Γ(aq,l)Γ(bq,l) Γ(aq,l +b q,l)Γ(a0 q,l)Γ(b0 q,l) } − nX i=1 GX g=1 ˜zi,gln˜zi,g, wheren g =n 0 g + Pn i=1 Ci,g,a g,k =a 0 g,k + Pn i=1 P j̸=i Ci,gCj,kYi,j, and bg,k =b ...

2025
[2015]

or continuous data (Lomet, 2012). For many versions of the LBM, Gibbs sampling is possible and the collapsed distributions p(C, C′|G, K) andp(Y|G, K, C, C′) can also be derived; see, e.g., Wyse and Friel (2012); Lomet (2012); Keribin et al. (2015). Thus, the THAMES can be defi...

2012
[2021]

76 Metodiev et al

is defined as the logarithm of the joint posterior density of the data and the allocation vector matrix ICL(C, G) = log(p(Y, C|G)) = log(p(Y|G, C)) + log(p(C|G)). 76 Metodiev et al. cluster1 2 34 5 6 7 8910 11 12 13 size3 1 522 41 79 20 50948 236 117 40 cluster 14 15 16 17 18 ...

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Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

[1] [1]

Abbe, E. (2018). Community detection and stochastic block models: recent developments.Journal of Machine Learning Research 18(177), 1–86. Agrawal, S. (2013). Mini-batch-k-means. R package.https://github.com/siddharth-agrawal/ Mini-Batch-K-Means. Aicher, C., A. Z. Jacobs, and A. Clauset (2015, June). Learning Latent Block Structure in Weighted Networks.Jou...

arXiv 2018

[2] [2]

This probability is much smaller than the usual probabilities obtained when simulating datasets of such a small size, which were often larger than 50 percent, in our experience. Supplement D: defining the THAMES for the latent block model The latent block model (LBM) has a very similar structure to the SBM, with the difference that the size of the rowsnma...

2010

[3] [3]

They are quite variable, with cluster 5, a cluster of size 6222, being the largest cluster, and cluster 11, a cluster of size 5, being the smallest cluster. ˆCM APcan be used to obtain estimates of the SBM parametersµand τby maximising thecomplete-data loglikelihood log(p(Y, ˆCM AP|G, τ, µ)) = nX i=1 GX g=1 ˆCM AP i,g log(τg) + nX i=1 GX g=1 X j̸=i GX k=1...

2008

[4] [5]

Brault, G

Keribin, C., V. Brault, G. Celeux, and G. Govaert (2015, November). Estimation and selection for the latent block model on categorical data.Statistics and Computing 25(6), 1201–1216. Latouche, P., E. Birmel´ e, and C. Ambroise (2014). Model selection in overlapping stochastic block models.Electronic Journal of Statistics 8(1), 762 –

2015

[5] [9]

, C(T) is an ergodic Markov chain with limiting distribution p(C|G, Y), the posterior distribution ofCgivenYandG, then ˆZ−1 RIS(G) almost surely converges toZ −1(G)

IfC (1), . . . , C(T) is an ergodic Markov chain with limiting distribution p(C|G, Y), the posterior distribution ofCgivenYandG, then ˆZ−1 RIS(G) almost surely converges toZ −1(G). IfC (1), . . . , C(T) is additionally ergodic of degree 2, then √ T( ˆZ−1 RIS(G)−Z −1(G))converges in distribution to a centred normal distribution. ProofFollows directly from ...

1994

[6] [12]

The labels of the nodes show the username of the user with the highest number of followers (taken as the maximum of this number over all timepoints on which the user sent a post within this dataset). ”Al Gore” represents cluster 11, whose central function as the core of the dataset is quite clear, as all other clusters are quoting or reposting its users w...

2023

[7] [13]

72 Metodiev et al. A topic-specific summary was obtained as follows: first, an estimate ˆβ is computed via the topicmodels package, where each entry ˆβ˜i,˜j is a matrix that gives the probability that a word with index ˜iappears given the topic with index ˜j(Gr¨ un and Hornik, 2011). Then, the entropy ˆβ˜i,˜j X i′ ˆβi′,˜j log( ˆβi′,˜j)/log( ˜K) +β ˜i,˜j i...

2011

[8] [14]

This may well be explained by topic 5, since this topic is associated with a specific movement, the #SaveSoil movement

This is particularly interesting because cluster 1 is the closest to the sociological definition of a community, with a fairly high internal connection probability. This may well be explained by topic 5, since this topic is associated with a specific movement, the #SaveSoil movement. This movement is also detected when using a different model choice crite...

2002