CLAIM C4 of entrylib:2026.000003 The observed-data likelihood of ADAM and iETS with unobserved periods
licence CC-BY-4.0; contributions C4

DEFINITIONS
D2 [concept] unobserved period: A period $t$ at which the potential value $z_t$ of the modelled variable is not observed ($O_t=0$): a period with zero demand in an intermittent demand model, where the demand size is not observed, or a period with a missing value.
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: $$L(\theta)=\int f(\mathbf y_{\mathrm{obs}},\mathbf z_{\mathrm{mis}};\theta)\,d\mathbf z_{\mathrm{mis}}.$$
D8 [method] SSOE Kalman filter: The recursion for the conditional mean $\mathbf a_t$ and scaled covariance $\mathbf P_t$ of the state of a linear single source of error model: at an observed period $$\begin{aligned} v_t &= z_t-\mathbf w'\mathbf a_t \\ f_t &= \mathbf w'\mathbf P_t\mathbf w+1 \\ \mathbf k_t &= (\mathbf F\mathbf P_t\mathbf w+\mathbf g)/f_t \\ \mathbf a_{t+1} &= \mathbf F\mathbf a_t+\mathbf k_t v_t \\ \mathbf P_{t+1} &= \mathbf F\mathbf P_t\mathbf F'+\mathbf g\mathbf g'-\mathbf k_t\mathbf k_t'f_t; \end{aligned}$$ at an unobserved period $$\begin{aligned} \mathbf a_{t+1} &= \mathbf F\mathbf a_t \\ \mathbf P_{t+1} &= \mathbf F\mathbf P_t\mathbf F'+\mathbf g\mathbf g'; \end{aligned}$$ started from $\mathbf a_1=\mathbf x_0$, $\mathbf P_1=\mathbf 0$.

CLAIMS
C4 [formal; principal; extension; support: formal_proof E4; assumes C30, C33, C34, C35; uses C1; verification agent_checked; verifiability open]
  For a pure additive single source of error model with normal errors and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] gives $z_t\mid Y_{t-1}\sim\mathcal N(\mathbf w'\mathbf a_t,\sigma^2 f_t)$ at the observed periods, the exact observed-data log-likelihood $$-\frac12\sum_{O_t=1}\left[\log(2\pi\sigma^2f_t)+\frac{v_t^2}{\sigma^2f_t}\right],$$ and the concentrated variance $$\hat\sigma^2=\frac{1}{T_1}\sum_{O_t=1}\frac{v_t^2}{f_t}.$$
  scope: population: pure additive single source of error models (ETS with additive components, ARIMA, regression) | data_type: univariate time series with unobserved periods | conditions: normally distributed errors; initial states treated as fixed parameters | excludes: models mixing additive and multiplicative components
  extends B6, at Kalman filter for state space models with correlated disturbances: The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated.
  extends B7, at Kalman filter for state space models with correlated disturbances: The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations.
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 $P(z_t=0)=0$.
C33 [descriptive; supporting; assumption (statistical, unchecked); verifiability not_applicable]
  Whether a period is observed carries no information about the unobserved value $z_t$ beyond the model, so that the missing values are missing at random.
C34 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The initial states of the single source of error model are fixed parameters, so that the state covariance is zero at the start of the sample.
C35 [descriptive; supporting; assumption (theorem_condition, unchecked); verifiability not_applicable]
  The errors of the pure additive model are normally distributed with constant variance $\sigma^2$.

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)
E4 [formal; foundation Kolmogorov probability axioms] Analytical derivation in the source draft (artefact A1).
  assumes C30, C33, C34, C35
  E4.1 $(\mathbf x_{t-1},z_t,\mathbf x_t)$ is jointly Gaussian given $Y_{t-1}$ (by C35, C30; linearity of Gaussian models)
  E4.2 $$\begin{aligned} \E(z_t\mid Y_{t-1}) &= \mathbf w'\mathbf a_t \\ \Var(z_t\mid Y_{t-1}) &= \sigma^2f_t \\ \Cov(\mathbf x_t,z_t\mid Y_{t-1}) &= \sigma^2(\mathbf F\mathbf P_t\mathbf w+\mathbf g) \end{aligned}$$ (by E4.1; moments of linear combinations)
  E4.3 Conditioning on $z_t$ gives the update; unobserved periods only propagate (by E4.2, C33; Gaussian conditioning)
  E4.4 The likelihood follows from the observed-data decomposition, and $\sigma^2$ concentrates out because $f_t$, $\mathbf k_t$ do not depend on it (by E4.3, C1; 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 C4, in: The observed-data likelihood of ADAM and iETS with unobserved periods. EntryLib. entrylib:2026.000003@v2#C4. https://entrylib.org/p/2026.000003/C4
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