Menu

Claim C2 · formal · entrylib:2026.000003@v2 · 2026-11-11

Corrected · 11 November 2026 at 11:22 UTC

The proofs of C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19 were checked by an independent agent on a second model (contributor P6; reports in artefacts A10 and A11). Its first round found five proofs that showed less than their claims stated; these claims are corrected to what is proved: C2 completes its argument with the normal-variance case, C5 states its condition as a nonzero $\mathbf w'\mathbf F^{i-1}\mathbf g$ and its conclusion for the objective and the variance, C10 states the conditions of the interval and that its bound is a quantile of the size distribution, C14 states the derivation of the sign (the sign printed by Svetunkov and Boylan, 2023, is in its relation), and C19 is stated for ETS(M,N,N) at horizons of two or more. The second round confirmed all ten in full. C14 rests on a formal proof (new evidence E15, definition D32) instead of an argument; D4 defines the term of each zero without presupposing its sign. Smaller precisions: C1 (positive sizes), C6 (the error mean enters the propagation through g), C8 depends on C15, C16 (alpha nonzero). Relations name references (B5-B7 added), and C1, C3, C6, C7, C9, C10 and C11 relate to the earlier work they extend or supersede. No claim ID changed.

By Ivan Svetunkov (ORCID 0000-0001-7826-0281)

The observed-data likelihood of ADAM and iETS with unobserved periods · C2

Principal claim C2 of this claimset (Svetunkov, I. (guarantor), 2026), with what it rests on. Read the whole claimset.

Copies a message for ChatGPT, Claude or Gemini: paste it there.

The message

Read https://entrylib.org/p/2026.000003/C2, one claim of a research paper with the evidence it rests on. Tell me briefly what it says and how well it is supported. If you cannot open the link, tell me instead of guessing. Then wait for my questions.

The claim

95C2 · formal · refutes earlier work · novelty: refutes earlier work, by the authors · no close match on EntryLib

The stationary points of the entropy-augmented likelihood of iETS differ in general from those of the observed-data likelihood: at a fixed point of the EM algorithm the gradient of the expected complete-data log-likelihood equals the score of the observed-data likelihood, while the term −Hz(θ)-H_z(\theta) of each period with zero demand has a gradient that is not zero in general, and for normal sizes with fixed location the two objectives are stationary at different variances.

Scope: population: iETS estimated by the entropy-augmented likelihood; data: intermittent demand time series; conditions: sizes unobserved at the periods with zero demand · Proof: E2 · Cited: Dempster et al. (1977) · Uses: C1 · Assumes: C31, C32 · Contradicts: Svetunkov and Boylan (2023), Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm) · Verification: checked by an agent

95C1 · formal · novelty: new, by the authors · no close match on EntryLib

For a model of intermittent demand yt=otzty_t=o_tz_t with Bernoulli occurrence ot∼Bernoulli(pt)o_t\sim\text{Bernoulli}(p_t) independent of a positive continuous size ztz_t, the observed-data likelihood is ∑Ot=1log⁡f(zt∣Yt−1;θ)+∑Ot=1log⁡pt+∑Ot=0log⁡(1−pt),\sum_{O_t=1}\log f(z_t\mid Y_{t-1};\theta) + \sum_{O_t=1}\log p_t + \sum_{O_t=0}\log(1-p_t), in which the unobserved sizes do not appear and the size and occurrence parts separate.

Scope: population: models of intermittent demand yt=otzty_t=o_tz_t (iETS, ADAM with an occurrence part); data: intermittent demand time series; conditions: Bernoulli occurrence independent of a continuous demand size · Proof: E1 · Assumes: C30, C31, C32 · Extends: Dempster et al. (1977) · Extends: Svetunkov and Boylan (2023), Appendix B, eq. B.6 · Verification: checked by an agent

Evidence

E1 · formal · cited by C1

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C31, C32

  1. P(yt=0∣Yt−1)=(1−pt)∫f(z∣Yt−1)dz=1−ptP(y_t=0\mid Y_{t-1})=(1-p_t)\int f(z\mid Y_{t-1})dz=1-p_t (independence and continuity; from C31, C32)
  2. yt>0y_t>0 has density ptf(yt∣Yt−1)p_tf(y_t\mid Y_{t-1}) (independence; from C31)
  3. The joint density of the observed data is the product of these conditional terms (prediction-error decomposition; from E1.1, E1.2)

E2 · formal · cited by C2

Analytical derivation in the source draft (artefact A1).

Foundation: Kolmogorov probability axioms · Assumes: C31, C32

  1. ∇θQ(θ∣θ(k))|θ=θ(k)=∇ℓ(θ(k)),\nabla_\theta Q(\theta\mid\theta^{(k)})\big|_{\theta=\theta^{(k)}}=\nabla\ell(\theta^{(k)}), with QQ the expected complete-data log-likelihood given the observed data at θ(k)\theta^{(k)} and ℓ\ell the observed-data likelihood (Fisher's identity (differentiation under the integral); from B1, C1)
  2. A fixed point of EM is a stationary point of ℓ\ell (EM identity; from E2.1)
  3. For normal sizes with fixed location, ℓ(σ2)=−T12log⁡σ2−S2σ2+c,S=∑Ot=1et2,\ell(\sigma^2)=-\frac{T_1}{2}\log\sigma^2-\frac{S}{2\sigma^2}+c,\qquad S=\sum_{O_t=1}e_t^2, stationary at σ2=S/T1\sigma^2=S/T_1 (differentiation; from C1, D3)
  4. Each zero adds ∫fzlog⁡fzdz=−12log⁡(2πeσ2)\int f_z\log f_z\,dz=-\frac12\log(2\pi e\sigma^2), whose gradient in σ2\sigma^2 is −1/(2σ2)≠0-1/(2\sigma^2)\neq0; so the entropy-augmented likelihood is ℓH(σ2)=ℓ(σ2)−T02log⁡(2πeσ2),\ell_H(\sigma^2)=\ell(\sigma^2)-\frac{T_0}{2}\log(2\pi e\sigma^2), stationary at σ2=S/T\sigma^2=S/T with T=T1+T0T=T_1+T_0 (normal entropy and differentiation; from D4, D32, E2.3)
  5. S/T≠S/T1S/T\neq S/T_1 whenever T0>0T_0>0 and S>0S>0, so the stationary points differ (comparison; from E2.3, E2.4)

Assumptions

C30 · statistical · unchecked

The errors ϵt\epsilon_t of the single source of error model are independent across periods. Rested on by C1.

C31 · theorem_condition · unchecked

In the intermittent demand model the occurrence oto_t is independent of the potential demand size ztz_t. Rested on by C2, C1, E1, E2.

C32 · theorem_condition · unchecked

The potential demand size ztz_t has a continuous distribution, so that P(zt=0)=0P(z_t=0)=0. Rested on by C2, C1, E1, E2.

Definitions

Concepts 3

D3 observed-data likelihood: The joint density of the observed data only, obtained by integrating the complete-data likelihood over the values 𝐳mis\mathbf z_{\mathrm{mis}} at the unobserved periods: L(θ)=∫f(𝐲obs,𝐳mis;θ)d𝐳mis.L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.

D4 entropy-augmented likelihood: The objective of the iETS estimator of Svetunkov and Boylan (2023, eq. B.6): the sum of the one-step log-densities of the observed demand sizes, plus, for each period with zero demand, the expected log-density of the unobserved size under its own distribution, ∫fzlog⁡fzdz\int f_z\log f_z\,dz, plus the Bernoulli log-likelihood of the occurrence. Source: Svetunkov and Boylan (2023).

D32 differential entropy: For a size with density fzf_z, Hz=−E⁡log⁡fz=−∫fz(z)log⁡fz(z)dz.H_z = -\E\log f_z = -\int f_z(z)\log f_z(z)\,dz.

Context

Contradicts Svetunkov and Boylan (2023), Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm) · own earlier work · from C2
Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand.
Extends Dempster et al. (1977) · from C1
Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood.
Extends Svetunkov and Boylan (2023), Appendix B, eq. B.6 · own earlier work · from C1
Svetunkov and Boylan define iETS, intermittent demand as the product of a Bernoulli occurrence and a demand size modelled by exponential smoothing, and estimate it with a likelihood that adds the differential entropy of the size for each period with zero demand.

References

  1. Dempster et al. (1977) Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum Likelihood from Incomplete Data Via the EM Algorithm. Journal of the Royal Statistical Society Series B: Statistical Methodology, 39(1), 1–22. doi:10.1111/j.2517-6161.1977.tb01600.x Used by: C2, Context
  2. Svetunkov and Boylan (2023) Svetunkov, I., & Boylan, J. E. (2023). iETS: State space model for intermittent demand forecasting. International Journal of Production Economics, 265, 109013. doi:10.1016/j.ijpe.2023.109013 Used by: Context, D4

Artefacts

Score

Scores are computed by rules from the claimset file (EntryLib rubric 0.2): they check the record, not the science, and may change as the rubric and reviews develop. The rubric.

C2: 95, High

Novelty: refutes earlier work, by the authors · no close match on EntryLib

  1. −3 Open problems Assumption C31 is not checked: In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  2. −1 Open problems Assumption C30 is not checked: The errors $\epsilon_t$ of the single source of error model are independent across periods. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  3. −1 Open problems Assumption C32 is not checked: The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.

C1: 95, High

Novelty: new, by the authors · no close match on EntryLib

  1. −3 Open problems Assumption C30 is not checked: The errors $\epsilon_t$ of the single source of error model are independent across periods. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  2. −1 Open problems Assumption C31 is not checked: In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.
  3. −1 Open problems Assumption C32 is not checked: The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$. To remove it: Test the assumption, or analyse how sensitive the claim is to it.