{"id":"4a3cb408-7bd0-4d22-b767-81501f9f3504","arxiv_id":"2607.23086","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The N-EM algorithm is the standard EM/MM lower-bound construction applied to integral likelihoods, repackaged with a 'normalization step'; it does not solve problems EM cannot solve.","lead":"This paper presents a 'normalized EM' algorithm that turns any likelihood with an integral into an EM-style maximization problem by normalizing the integrand. It is a useful recipe, but it is mathematically the same as the classical EM algorithm, and several of the worked examples contain errors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central novelty claim collapses: N-EM is exactly EM with the integration variable s treated as missing data, so it cannot solve problems EM cannot. Additionally, the printed Q1 in Eq. 3.29 violates the tangency condition Q(θt|θt)=ℓ(θt).","rationale":"The reader's verdict is REJECT with high correctness risk, and I agree that rejection is warranted. My principal concern differs from the reader's formal 'weakest assumption': the reader emphasizes unverified convergence conditions, while I find the more fundamental issue to be that the N-EM algorithm is mathematically identical to ordinary EM where the integration variable is the missing data. This undercuts the paper's central claim of novelty and of solving problems EM cannot. The paper itself proves EM is a special case (Section 2.4), which is logically inconsistent with the assertion of solving EM-impossible problems. The N-EM framework is a clean exposition of the standard EM lower-bound construction, but it is not a new algorithm. The reader also noted this overclaim in the rationale, so my agreement is partial rather than full. I also identified a separate internal problem in Eq. 3.29: as printed, the skew-normal Q-function violates the tangency condition (2.7), so the ascent property fails for that example. This may be a typesetting error, but under the rule of treating the manuscript text as in-scope, it is a concrete defect that supports rejection of the submitted version. The proposed convergence theorems are standard MM results and are not machine-checked; the assumptions are merely asserted for the examples. All of these issues together justify maintaining the REJECT verdict. I would not change the reader's verdict, hence UNCHANGED.","tokens_in":26513,"tokens_out":16399,"duration_ms":137373,"concrete_test":"For the gamma-integral model of Section 3.1, form the complete-data density f(x_i,s|θ)=s^{θ-2}e^{-s}I(s>x_i)/Γ(θ), treat s as missing data, and derive the EM Q-function (2.16). If it equals Eq. (3.25), then N-EM is exactly EM and the claim of solving a problem EM cannot is refuted. Separately, plug θ=θ^{(t)} into Eq. (3.29) and compare with ℓ1(θ^{(t)}|Y); the printed expression will differ, showing the tangency condition fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central assertion is that N-EM solves problems EM cannot. But the N-EM construction is identical to standard EM applied with the integration variable as latent data. For likelihood (2.1), define complete-data density f(x_i,s|θ)=c(x_i,θ)h(s|x_i,θ). Then ∫ f(x_i,s|θ)ds=f(x_i|θ). EM's E-step uses the conditional density f(s|x_i,θ_t)=h(s|x_i,θ_t)/∫h(s'|x_i,θ_t)ds', which is exactly the N-step ndf (2.3), and EM's Q-function (2.16) equals N-EM's Q (2.4). Section 2.4 concedes EM is a special case. Therefore every 'solved for the first time' example in Section 3 is solved by ordinary EM with latent variable s. The claim that N-EM can solve problems EM cannot is thus false. This does not make the numerical algorithm fail, but it invalidates the paper's stated contribution. In addition, the printed Q1 in Eq. 3.29 does not satisfy the tangency condition (2.7): evaluating at θ=θ^{(t)} does not reduce to ℓ1(θ^{(t)}|Y) because the term -nσ_*^{2(t)} lacks the denominator 2σ_*^2 and the quadratic term n(μ-μ^{(t)})^2/(2σ_*^2) has the wrong sign. As printed, the ascent property for the skew-normal update does not hold.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an 'N-EM' algorithm for maximum likelihood estimation when the likelihood has the form f(x;θ)=c(x,θ)∫_S h(s|x,θ)ds. At iteration t one forms the normalized density g(s|x_i,θ^(t))=h(s|x_i,θ^(t))/∫_S h(s'|x_i,θ^(t))ds', constructs Q(θ|θ^(t))=∑_i log c(x_i,θ)+∑_i E_g log h(s|x_i,θ), and maximizes Q. The paper proves an ascent property via Jensen's inequality, argues that this choice of g is optimal among a class of lower-bound densities, and applies the recipe to several models: gamma-integral, skew normal-normal, skew t-normal, skew Laplace-normal, two-sided truncated normal, multivariate Laplace-type models, PIG data, and grouped data. Numerical experiments and convergence results are presented.","tokens_in":26900,"tokens_out":13950,"duration_ms":131524,"significance":"The core lower-bound derivation in Section 2.2 is correct and self-contained: Jensen's inequality produces a minorizing function, and the normalized g at θ^(t) achieves tangency, so the ascent property holds for the generic construction. The framework is a legitimate pedagogical unification of EM-type constructions. However, the construction is mathematically identical to ordinary EM in which the integration variable s is treated as the latent variable, so the paper's central selling point—that N-EM solves problems EM cannot—is not valid. In addition, the displayed surrogate Q1 in Eq. (3.29) contains sign errors that violate the tangency condition, so one flagship application does not satisfy even the basic ascent property. The numerical experiments are extensive but do not compare with baseline EM/MM algorithms, and the convergence claims rest on unverified regularity assumptions.","major_comments":[{"comment":"The central claim that N-EM solves problems EM cannot is contradicted by the construction itself. For likelihood (2.1), define the complete-data density f(x,s;θ)=c(x,θ)h(s|x,θ). Then ∫_S f(x,s;θ)ds=f(x;θ), and the conditional density is f(s|x,θ^(t))=h(s|x,θ^(t))/∫_S h(s'|x,θ^(t))ds', which is exactly the N-step ndf (2.3). The Q-function in (2.4) is then exactly the standard EM Q-function (2.16) for this complete-data model. Thus every 'first time' application in Section 3 is an ordinary EM algorithm with latent variable s, and the abstract's assertion that N-EM solves problems 'which cannot be addressed by the EM algorithm' is unsupported. This invalidates the paper's stated principal contribution.","section":"Sections 1, 2.1, 2.4, and 3"},{"comment":"Equation (3.29) does not satisfy the tangency condition (2.7). Evaluating Q1 at θ=θ^(t) does not reproduce ℓ1: the term -nσ_*^{2(t)} lacks the denominator 2σ_*²; the term n(μ-μ^(t))²/(2σ_*²) has the wrong sign; and the boundary correction term should involve (x_i-μ^(t)) rather than (x_i+μ^(t)-2μ), with additional σ_*^{4(t)}g_1^2 terms from the truncated-normal variance. The exact truncated-normal expectation gives -nσ_*^{2(t)}/(2σ_*²) - n(μ-μ^(t))²/(2σ_*²) + (1/(2σ_*²))∑_i σ_*^{2(t)}(x_i-μ^(t))g_1(...) + (1/(2σ_*²))∑_i σ_*^{4(t)}g_1(...)^2. As printed, Q1 is not tangent to ℓ1, so the monotone ascent property (2.9) is not guaranteed for the SNND update (3.30), and the numerical results for SNND in Table 1 are not backed by the stated theory.","section":"Section 3.2.1, Eq. (3.29)"},{"comment":"The convergence results are conditional on unverified hypotheses. The manuscript asserts that the surrogate functions in (3.29), (3.34), (3.39), (3.43), (4.45), (4.52), (4.58) and (4.55) are strictly concave, that -ℓ is coercive, and that stationary points are isolated, but no verification is supplied for any of the eight examples. For instance, for the skew-normal surrogate, even after correcting (3.29), global strict concavity in (μ,σ²,δ) through σ_*=σ/δ is not established; for the skew-t surrogate, the weights w_i^(t) and the log terms make strict concavity similarly non-obvious. The statement that 'the local convergence results follow directly by Proposition 1' is therefore not supported by the manuscript.","section":"Section 6, Propositions 1–2 and Corollaries 1–2"},{"comment":"The numerical experiments contain no comparison with existing EM algorithms, MM algorithms, or numerical quadrature/Newton-type methods for the same problems. Consequently, the claims that N-EM has 'superiority in relieving computational burden' or is efficient are not demonstrated. The simulations show convergence of the proposed updates, but they do not establish any advantage over existing methods, especially given that the updates are standard EM updates for the same latent-variable representations.","section":"Section 5 (empirical comparisons)"}],"minor_comments":[{"comment":"Section numbering is inconsistent: the applications in Section 3 are also referred to as Section 4 in the numerical section, and the finite homogeneous summation promised in the abstract and introduction does not appear as a developed section. The folded normal distribution (FND) is mentioned in Section 5 but not derived in the provided text.","section":"General organization"},{"comment":"There are numerous typos: 'constrcuting' (Section 1), 'calcualte' (Section 4.1.1), 'Tpye' (Table 2), 'in of' (Section 4.1.3), and 'regraded' (Section 2.4). These should be corrected.","section":"Typos"},{"comment":"The constants c_0^(t), c_01^(t), c_02^(t) in inequalities (2.20)–(2.22) are not explicitly defined. This makes the display harder to check; please provide their definitions.","section":"Section 2.5.2"}],"recommendation":"reject","confidential_remarks":"The main obstacle is the novelty claim: as the construction is equivalent to EM with the integration variable as latent data, the paper's central contribution is not as stated. The sign error in Eq. (3.29) compounds the problem because a flagship example does not satisfy the paper's own ascent condition. A repositioning as a didactic/unified perspective on EM might be possible, but that is not what the manuscript currently claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe punchline: N-EM is EM with the integration variable treated as missing data. The paper's repeated claim that it solves problems EM cannot is false, and they even prove EM is a special case in §2.4. What is left is a clean recipe for deriving EM updates for integral likelihoods, plus some M-step updates for specific models that may not have appeared before.\n\nThe core math in §2 is sound: Jensen gives a minorizing function, and optimizing the lower bound over the auxiliary density returns the conditional density of the latent integration variable. That is textbook EM/MM material (Lange 2000; Neal & Hinton 1998, not cited), but the Lagrange derivation is clearly written and the examples are worked out in detail. For that reason the paper is useful as a tutorial-style unification.\n\nSoft spots, in order of seriousness:\n\n1. Novelty overclaim. Any likelihood of the form (2.1) admits a missing-data structure by taking s as the latent variable. So every \"first time\" example in §3 is just a standard EM derivation with a particular latent variable. That doesn't make the updates wrong, but it collapses the paper's main contribution.\n\n2. Eq. (3.29) has sign errors in the printed Q1: the correct truncated-normal expectation gives −n(μ−μ^(t))^2/(2σ_*^2), not plus, and the σ_*^{2(t)} term needs a denominator 2σ_*^2. The M-step updates in (3.30) look like they were derived from the corrected version, so this is probably a typesetting slip, but as printed the equation cannot be the Q-function the algorithm maximizes. The stress-test claim that Q itself must satisfy the tangency condition is a misunderstanding — Q is not the minorizer, Q* is.\n\n3. The convergence section asserts strict concavity and differentiability of the surrogate functions for all eight examples with no verification. For several of those models the Q is not jointly concave, so Corollaries 1–2 overclaim.\n\n4. Tables 2 and 3 report simulations for LMM and FND, but no N-EM algorithm for either model appears anywhere in the paper. That is a missing-methods problem.\n\nThe exposition also needs heavy copyediting; there are typos and broken sentences throughout.\n\nWho should read it: someone teaching EM/data augmentation and wanting a case-study collection. As a research contribution it needs major restructuring: drop the \"cannot solve\" language, fix the equations, prove or qualify the convergence statements, and either provide the LMM/FND derivations or remove those tables. I'd send it to referees rather than desk-reject, because the specific updates are potentially citeable and the errors are fixable, but I'd expect a report that asks for very substantial revision.","headline":"N-EM is a repackaging of EM with the integration variable as missing data; the specific updates look useful, but the paper's central novelty claim and several technical details need major correction.","tokens_in":27381,"tokens_out":9164,"would_cite":false,"duration_ms":75779,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F10","62-08"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new three-step N-EM algorithm maximizes likelihoods containing integrals without hand-crafted latent variables, and it subsumes classic EM as a special case.","keywords":["N-EM algorithm","normalized density function","EM algorithm","MM algorithm","Jensen's inequality","optimal lower bound","maximum likelihood estimation","integral likelihood"],"falsifier":"Run the N-EM updates in (3.30) on a dataset drawn from the skew-normal-normal model and record the log-likelihood; if any iteration decreases the log-likelihood (beyond numerical tolerance), the ascent property is violated. Alternatively, numerically check strict concavity of Q(θ|θ^(t)) at a non-stationary point for the truncated-normal example; if a direction of zero curvature exists, Proposition 1's rate claim is void.","tokens_in":26413,"feed_emoji":"📊","tokens_out":4208,"duration_ms":38588,"temperature":0.7,"pith_summary":"The paper proposes the normalized expectation-maximization (N-EM) algorithm, a three-step recipe for maximum likelihood estimation when the log-likelihood contains an integral term or a finite homogeneous summation. Instead of searching for an artful missing-data structure case by case, the user normalizes the integrand to form a density, builds a Q-function via expectation under that density, and maximizes as in standard EM. The paper proves the algorithm has the ascent property, that its chosen normalized density gives the optimal lower bound, and that classic EM arises as a special case. It then derives explicit N-EM updates for eight models, including several whose MLEs had not previously been computed by EM-type algorithms, and reports simulations supporting monotone convergence.","feed_headline":"N-EM algorithm fits integral-likelihood models, no latent variables","feed_subtitle":"A single normalization step builds the surrogate that classic EM gets only by hand-crafted data augmentation.","key_machinery":"The normalized density function (ndf): g(s|x,θ)=h(s|x,θ)/∫_S h(s'|x,θ) ds'. It converts the integral term into an expectation under a proper probability density, makes Jensen's inequality tight at the current iterate, and thereby turns maximum likelihood into the EM-style iteration θ^(t+1)=argmax_θ Q(θ|θ^(t)). It carries the whole argument: normalization is what manufactures the missing-data-like structure automatically.","core_discovery":"For likelihoods of the form f(x;θ)=c(x,θ)∫_S h(s|x,θ)ds with h≥0 and finite integral, the paper shows that normalizing h by its integral at the current parameter estimate yields a density g; using this g as the mixing measure makes Q(θ|θ^(t)) = Σ log c(x_i,θ) + Σ ∫ log h(s|x_i,θ) g(s|x_i,θ^(t)) ds a surrogate that lies below the log-likelihood and touches it at θ=θ^(t). Maximizing this Q therefore ascends the likelihood monotonically. The paper further shows this g is the best possible choice among all candidate densities, in the sense that it produces the optimal lower bound, and that the classical EM surrogate is recovered when the integrand is a complete-data joint density.","pith_inferences":["A natural test of the framework is whether the N-step can be applied to likelihoods whose integral is only computable by Monte Carlo; the paper does not develop this, but the surrogate construction would remain valid with an estimated g.","Because the paper's optimality argument for the ndf is variational, the same Lagrange-multiplier device could be adapted to choose not only g but also data-dependent transformations of the integrand, potentially tightening the lower bound further.","The N-EM view reframes EM's 'art' as a normalization step; if that view is right, it suggests textbook EM derivations can be taught as a mechanical integral-normalization exercise rather than as case-by-case tricks.","For models with multiple or nested integral terms, the N-step could be applied componentwise; the paper does not explore this, but the convexity argument appears to carry through."],"forward_implications":["For any likelihood of the displayed integral form, an N-EM algorithm can be written down mechanically from the integrand, without designing latent variables.","Classical EM surrogates are a special case: the conditional predictive density of the missing data is exactly the ndf of the complete-data joint density.","The ascent property guarantees the observed log-likelihood never decreases across iterations, so the iteration is monotone under standard compactness/continuity conditions.","The same framework handles finite homogeneous sums, which are discrete integrals, so models whose likelihood contains such sums also fit the recipe.","If the local convergence conditions hold, the iteration converges linearly to a local maximizer with rate given by the spectral radius of the algorithm's map."],"fun_headline_variants":["N-EM: EM without the latent-variable headache","Normalized EM handles integral likelihoods directly","New EM variant skips latent variables, still ascends","N-EM: one normalization step replaces data augmentation"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The convergence theorems rely on the surrogate Q-functions being strictly concave and differentiable, the negative log-likelihood being coercive, and stationary points being isolated; the paper asserts these hold for its examples without verifying them, and if any Q lacks strict concavity the claimed linear convergence does not follow.","fun_headline_variants_meta":{"raw":{"variants":["N-EM: EM without the latent-variable headache","Normalized EM handles integral likelihoods directly","New EM variant skips latent variables, still ascends","N-EM: one normalization step replaces data augmentation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1303,"prompt_tokens":844,"completion_tokens":459,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":396}},"tokens_in":588,"tokens_out":459,"duration_ms":5331,"temperature":1.0,"reasoning_tokens":396,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:38:10.367197+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the N-EM updates in (3.30) on a dataset drawn from the skew-normal-normal model and record the log-likelihood; if any iteration decreases the log-likelihood (beyond numerical tolerance), the ascent property is violated. Alternatively, numerically check strict concavity of Q(θ|θ^(t)) at a non-stationary point for the truncated-normal example; if a direction of zero curvature exists, Proposition 1's rate claim is void.","supporting_citations":[],"review_version":1}