CLAIM C1 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C1

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: $$L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$$

CLAIMS
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)

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.

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C1, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C1. https://entrylib.org/p/2026.000003/C1
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