CLAIM C2 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C2

DEFINITIONS
D3 [concept] observed-data likelihood: The joint density of the observed data only, obtained by integrating the complete-data likelihood over the values $\mathbf z_{\mathrm{mis}}$ at the unobserved periods: $$L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$$
D4 [concept] 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, $\int f_z\log f_z\,dz$, plus the Bernoulli log-likelihood of the occurrence. [source B4]
D32 [concept] differential entropy: For a size with density $f_z$, $$H_z = -\E\log f_z = -\int f_z(z)\log f_z(z)\,dz.$$

CLAIMS
C2 [formal; principal; refutation; support: formal_proof E2, cited B1; assumes C31, C32; uses C1; verification agent_checked; verifiability open]
  The stationary points of the [[D4:entropy-augmented likelihood]] of iETS differ in general from those of the [[D3: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 $-H_z(\theta)$ 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_type: intermittent demand time series | conditions: sizes unobserved at the periods with zero demand
  contradicts B4, at Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm), own earlier work: 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.
C1 [formal; principal; support: formal_proof E1; assumes C30, C31, C32; verification agent_checked; verifiability open]
  For a model of intermittent demand $y_t=o_tz_t$ with Bernoulli occurrence $o_t\sim\text{Bernoulli}(p_t)$ independent of a positive continuous size $z_t$, the [[D3:observed-data likelihood]] is $$\sum_{O_t=1}\log f(z_t\mid Y_{t-1};\theta) + \sum_{O_t=1}\log p_t + \sum_{O_t=0}\log(1-p_t),$$ in which the unobserved sizes do not appear and the size and occurrence parts separate.
  scope: population: models of intermittent demand $y_t=o_tz_t$ (iETS, ADAM with an occurrence part) | data_type: intermittent demand time series | conditions: Bernoulli occurrence independent of a continuous demand size
  extends B1: Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood.
  extends B4, at Appendix B, eq. B.6, own earlier work: 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.
C30 [descriptive; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  The errors $\epsilon_t$ of the single source of error model are independent across periods.
C31 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$.
C32 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$.

EVIDENCE
E1 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E1.1 $P(y_t=0\mid Y_{t-1})=(1-p_t)\int f(z\mid Y_{t-1})dz=1-p_t$ (by C31, C32; independence and continuity)
  E1.2 $y_t>0$ has density $p_tf(y_t\mid Y_{t-1})$ (by C31; independence)
  E1.3 The joint density of the observed data is the product of these conditional terms (by E1.1, E1.2; prediction-error decomposition)
E2 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C31, C32
  E2.1 $$\nabla_\theta Q(\theta\mid\theta^{(k)})\big|_{\theta=\theta^{(k)}}=\nabla\ell(\theta^{(k)}),$$ with $Q$ the expected complete-data log-likelihood given the observed data at $\theta^{(k)}$ and $\ell$ the [[D3:observed-data likelihood]] (by B1, C1; Fisher's identity (differentiation under the integral))
  E2.2 A fixed point of EM is a stationary point of $\ell$ (by E2.1; EM identity)
  E2.3 For normal sizes with fixed location, $$\ell(\sigma^2)=-\frac{T_1}{2}\log\sigma^2-\frac{S}{2\sigma^2}+c,\qquad S=\sum_{O_t=1}e_t^2,$$ stationary at $\sigma^2=S/T_1$ (by C1, D3; differentiation)
  E2.4 Each zero adds $\int f_z\log f_z\,dz=-\frac12\log(2\pi e\sigma^2)$, whose gradient in $\sigma^2$ is $-1/(2\sigma^2)\neq0$; so the [[D4:entropy-augmented likelihood]] is $$\ell_H(\sigma^2)=\ell(\sigma^2)-\frac{T_0}{2}\log(2\pi e\sigma^2),$$ stationary at $\sigma^2=S/T$ with $T=T_1+T_0$ (by D4, D32, E2.3; normal entropy and differentiation)
  E2.5 $S/T\neq S/T_1$ whenever $T_0>0$ and $S>0$, so the stationary points differ (by E2.3, E2.4; comparison)

ARTEFACTS
A1 [manuscript] LaTeX source of the draft 'Estimating exponential smoothing with unobserved periods: the observed-data likelihood of ADAM and iETS' (October 2026).
A10 [proof] Report of the first round of the independent proof check by the contributor P6 (2026-11-11): for C1, C2, C5, C6, C8, C10, C12, C14, C16 and C19, the re-derivation, the numerical spot checks, the outcome and the gaps found.
A11 [proof] Report of the second round of the independent proof check by P6 (2026-11-11), on C2, C5, C10, C14 and C19 as corrected after the first round, and on the smaller changes to C1, C6, C8 and C16.
A14 [code] Numerical spot checks of the first round of the proof check (Python with numpy); the report A10 says which claim each serves.
A16 [code] Numerical spot checks of the proof check (Python with numpy), used in both rounds; the reports A10 and A11 say which claim each serves.
A17 [code] Numerical spot checks of the second round of the proof check (Python with numpy); the report A11 says which claim each serves.

REFERENCES
B1 10.1111/j.2517-6161.1977.tb01600.x (doi)
B4 10.1016/j.ijpe.2023.109013 (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C2, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C2. https://entrylib.org/p/2026.000003/C2
The claimset it is from: Svetunkov, I. (guarantor) (2026). The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2. https://entrylib.org/p/2026.000003@v2