{
 "bundle_of": "C5",
 "schema_version": "0.41",
 "paper": {
  "id": "entrylib:2026.000003",
  "version": 2,
  "title": "The observed-data likelihood of ADAM and iETS with unobserved periods",
  "licence": "CC-BY-4.0",
  "status": "corrected",
  "date": "2026-11-11",
  "status_notice": {
   "date": "2026-11-11",
   "reason_code": "correction",
   "statement": "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.",
   "issued_by": "Ivan Svetunkov (ORCID 0000-0001-7826-0281)"
  }
 },
 "claims": [
  {
   "id": "C5",
   "statement": "For pure additive single source of error models with normal errors and [[D2:unobserved period]]s, if $c_i=\\mathbf w'\\mathbf F^{i-1}\\mathbf g\\neq0$ for some $i$ shorter than an interval, the [[D5:zero-error recursion]] likelihood is not the observed-data likelihood and its maximiser in the error variance is in general not the maximum likelihood estimate, because at the observed period after such an interval the recursion uses the error variance $\\sigma^2$ instead of $\\sigma^2f_t>\\sigma^2$.",
   "about": {
    "subject": "D5",
    "relation": "is not the maximum likelihood estimator of",
    "object": "pure additive SSOE models with normal errors",
    "comparator": null,
    "conditions": [
     "D2",
     "nonzero persistence vector"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "refutation",
   "scope": {
    "population": "pure additive single source of error models",
    "data_type": "univariate time series with unobserved periods",
    "conditions": [
     "normally distributed errors",
     "$\\mathbf w'\\mathbf F^{i-1}\\mathbf g\\neq0$ for some $i$ shorter than an interval"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E6"
    }
   ],
   "assumptions": [
    "C30",
    "C33",
    "C34",
    "C35"
   ],
   "depends_on": [
    "C4"
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "relations": [
    {
     "type": "contradicts",
     "reference": "B4",
     "locator": "Appendix C (estimation with zero errors at the unobserved periods)",
     "self": true,
     "note": "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."
    }
   ],
   "checks": [
    {
     "by": "P6",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "Re-derived f=1+sum c_i^2>1 for some c_i!=0 and that the zero-error likelihood uses sigma^2 where the exact one uses sigma^2 f_t, so the likelihoods differ; the variance maximisers (1/T1)sum e^2 and (1/T1)sum v^2/f differ because their difference is positive when the first observed period after a gap is the last observation. In an ETS(A,N,N) check the two sums were 7.4975 and 6.7015, and the difference was positive in all 2000 random samples. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)"
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C1",
   "statement": "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.",
   "about": {
    "subject": "D3",
    "relation": "separates into a size part and an occurrence part for",
    "object": "models of intermittent demand",
    "comparator": null,
    "conditions": [
     "Bernoulli occurrence independent of a continuous size"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "no_prior_found",
   "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"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E1"
    }
   ],
   "assumptions": [
    "C30",
    "C31",
    "C32"
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "relations": [
    {
     "type": "extends",
     "reference": "B1",
     "self": false,
     "note": "Dempster, Laird and Rubin define the EM algorithm, which maximises the likelihood of incomplete data through the expected complete-data log-likelihood."
    },
    {
     "type": "extends",
     "reference": "B4",
     "locator": "Appendix B, eq. B.6",
     "self": true,
     "note": "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."
    }
   ],
   "checks": [
    {
     "by": "P6",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "Re-derived the prediction-error factorisation: P(y=0|past)=1-p by independence and continuity, and y>0 has density p*f, so the log-likelihood separates into size and occurrence parts. Checked numerically on ETS(A,N,N) with two unobserved periods: the integral of the complete-data density over the unobserved sizes equals the Kalman predictive-density sum to 2e-13. (first round; report A10, scripts A14-A16.)"
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C4",
   "statement": "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 $$\\hat\\sigma^2=\\frac{1}{T_1}\\sum_{O_t=1}\\frac{v_t^2}{f_t}.$$",
   "about": {
    "subject": "D8",
    "relation": "gives the exact observed-data likelihood of",
    "object": "pure additive SSOE models with normal errors",
    "comparator": null,
    "conditions": [
     "D2"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "extension",
   "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"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E4"
    }
   ],
   "assumptions": [
    "C30",
    "C33",
    "C34",
    "C35"
   ],
   "depends_on": [
    "C1"
   ],
   "verification": "agent_checked",
   "checks": [
    {
     "by": "P5",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "The independent proof checker re-derived the proof of E4 step by step and confirmed it."
    }
   ],
   "provenance": "author_stated",
   "relations": [
    {
     "type": "extends",
     "reference": "B6",
     "locator": "Kalman filter for state space models with correlated disturbances",
     "self": false,
     "note": "The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated."
    },
    {
     "type": "extends",
     "reference": "B7",
     "locator": "Kalman filter for state space models with correlated disturbances",
     "self": false,
     "note": "The reference gives the Kalman filter for state space models whose measurement and transition disturbances are correlated, and its treatment of missing observations."
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C30",
   "statement": "The errors $\\epsilon_t$ of the single source of error model are independent across periods.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "statistical",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C31",
   "statement": "In the intermittent demand model the occurrence $o_t$ is independent of the potential demand size $z_t$.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "theorem_condition",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C32",
   "statement": "The potential demand size $z_t$ has a continuous distribution, so that $P(z_t=0)=0$.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "theorem_condition",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C33",
   "statement": "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.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "statistical",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C34",
   "statement": "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.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "theorem_condition",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C35",
   "statement": "The errors of the pure additive model are normally distributed with constant variance $\\sigma^2$.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "theorem_condition",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  }
 ],
 "definitions": [
  {
   "id": "D2",
   "term": "unobserved period",
   "definition": "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.",
   "kind": "concept"
  },
  {
   "id": "D3",
   "term": "observed-data likelihood",
   "definition": "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: $$L(\\theta)=\\int f(\\mathbf y_{\\mathrm{obs}},\\mathbf z_{\\mathrm{mis}};\\theta)\\,d\\mathbf z_{\\mathrm{mis}}.$$",
   "kind": "concept"
  },
  {
   "id": "D5",
   "term": "zero-error recursion",
   "definition": "The estimator used in ADAM and in iETS (Svetunkov and Boylan 2023, Appendix C): the single source of error recursion run with $\\epsilon_t=0$ at the unobserved periods and with the fixed persistence vector $\\mathbf g$ at the observed ones, with the likelihood taken as the product of one-step densities at the observed periods.",
   "kind": "method",
   "source": "B4"
  },
  {
   "id": "D8",
   "term": "SSOE Kalman filter",
   "definition": "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 $\\mathbf a_1=\\mathbf x_0$, $\\mathbf P_1=\\mathbf 0$.",
   "kind": "method"
  }
 ],
 "evidence": [
  {
   "id": "E1",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E1.1",
     "statement": "$P(y_t=0\\mid Y_{t-1})=(1-p_t)\\int f(z\\mid Y_{t-1})dz=1-p_t$",
     "by": [
      "C31",
      "C32"
     ],
     "rule": "independence and continuity"
    },
    {
     "id": "E1.2",
     "statement": "$y_t>0$ has density $p_tf(y_t\\mid Y_{t-1})$",
     "by": [
      "C31"
     ],
     "rule": "independence"
    },
    {
     "id": "E1.3",
     "statement": "The joint density of the observed data is the product of these conditional terms",
     "by": [
      "E1.1",
      "E1.2"
     ],
     "rule": "prediction-error decomposition"
    }
   ],
   "artefacts": [
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   "availability": {
    "level": "public"
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   "assumptions": [
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    "C32"
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  },
  {
   "id": "E4",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E4.1",
     "statement": "$(\\mathbf x_{t-1},z_t,\\mathbf x_t)$ is jointly Gaussian given $Y_{t-1}$",
     "by": [
      "C35",
      "C30"
     ],
     "rule": "linearity of Gaussian models"
    },
    {
     "id": "E4.2",
     "statement": "$$\\begin{aligned} \\E(z_t\\mid Y_{t-1}) &= \\mathbf w'\\mathbf a_t \\\\ \\Var(z_t\\mid Y_{t-1}) &= \\sigma^2f_t \\\\ \\Cov(\\mathbf x_t,z_t\\mid Y_{t-1}) &= \\sigma^2(\\mathbf F\\mathbf P_t\\mathbf w+\\mathbf g) \\end{aligned}$$",
     "by": [
      "E4.1"
     ],
     "rule": "moments of linear combinations"
    },
    {
     "id": "E4.3",
     "statement": "Conditioning on $z_t$ gives the update; unobserved periods only propagate",
     "by": [
      "E4.2",
      "C33"
     ],
     "rule": "Gaussian conditioning"
    },
    {
     "id": "E4.4",
     "statement": "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"
     ],
     "rule": "prediction-error decomposition"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C30",
    "C33",
    "C34",
    "C35"
   ]
  },
  {
   "id": "E6",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E6.1",
     "statement": "After $j-1$ unobserved periods from $\\mathbf P=\\mathbf 0$, $f=1+\\sum_{i<j}c_i^2$ with $c_i=\\mathbf w'\\mathbf F^{i-1}\\mathbf g$",
     "by": [
      "D8"
     ],
     "rule": "iteration of the propagation"
    },
    {
     "id": "E6.2",
     "statement": "$f_t>1$ when $c_i\\neq0$ for some $i<j$",
     "by": [
      "E6.1"
     ],
     "rule": "substitution"
    },
    {
     "id": "E6.3",
     "statement": "The zero-error recursion uses variance $\\sigma^2$ at every observed period, so its objective differs from the exact likelihood",
     "by": [
      "E6.2",
      "D5",
      "C4"
     ],
     "rule": "comparison"
    },
    {
     "id": "E6.4",
     "statement": "Both predict the first observed period after the first interval from the states propagated by $\\mathbf F$ with zero errors, so its prediction error $e_t$ is the same in both",
     "by": [
      "D5",
      "D8"
     ],
     "rule": "zero mean of the errors in the interval"
    },
    {
     "id": "E6.5",
     "statement": "For given other parameters the maximiser in $\\sigma^2$ is $\\frac{1}{T_1}\\sum e_t^2$ for the zero-error recursion and $\\frac{1}{T_1}\\sum v_t^2/f_t$ for the exact likelihood, which differ unless the data satisfy $\\sum e_t^2=\\sum v_t^2/f_t$, a term with $f_t>1$ and $v_t=e_t$ being present",
     "by": [
      "E6.2",
      "E6.4",
      "C4"
     ],
     "rule": "differentiation in $\\sigma^2$"
    }
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   "availability": {
    "level": "public"
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 "references": [
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   "id": "B4",
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 ]
}