{
 "bundle_of": "C3",
 "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": "C3",
   "statement": "For normally distributed demand sizes with fixed location, the maximiser in the variance of the [[D4:entropy-augmented likelihood]] is $\\frac{T_1}{T}\\hat\\sigma^2$, where $\\hat\\sigma^2=\\frac{1}{T_1}\\sum_{O_t=1}e_t^2$ is the fixed point of the EM algorithm, $T$ the number of periods and $T_1$ the number of periods with observed sizes.",
   "about": {
    "subject": "D4",
    "relation": "is maximised at T1/T times the EM fixed point in",
    "object": "the variance of the sizes",
    "comparator": null,
    "conditions": [
     "normal sizes",
     "location held fixed"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "no_prior_found",
   "scope": {
    "population": "iETS with normally distributed demand sizes",
    "conditions": [
     "location of the sizes held fixed",
     "variance as the only free parameter"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E3"
    }
   ],
   "depends_on": [
    "C2"
   ],
   "verification": "agent_checked",
   "checks": [
    {
     "by": "P5",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "The independent proof checker re-derived the proof of E3 step by step and confirmed it."
    }
   ],
   "provenance": "author_stated",
   "relations": [
    {
     "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."
    }
   ],
   "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": "C2",
   "statement": "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.",
   "about": {
    "subject": "D4",
    "relation": "has stationary points that differ in general from those of",
    "object": "D3",
    "comparator": null,
    "conditions": []
   },
   "type": "formal",
   "role": "principal",
   "novelty": "refutation",
   "scope": {
    "population": "iETS estimated by the entropy-augmented likelihood",
    "data_type": "intermittent demand time series",
    "conditions": [
     "sizes unobserved at the periods with zero demand"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E2"
    },
    {
     "kind": "cited",
     "reference": "B1",
     "self": false
    }
   ],
   "assumptions": [
    "C31",
    "C32"
   ],
   "depends_on": [
    "C1"
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "relations": [
    {
     "type": "contradicts",
     "reference": "B4",
     "locator": "Appendix B, eq. B.6 (likelihood with the differential entropy of the unobserved sizes, derived by analogy with the EM algorithm)",
     "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 Fisher's identity (gradient of Q at the current estimate equals the observed-data score, valid at every iterate and so at EM fixed points) and the normal-variance example: the observed-data likelihood is stationary at S/T1 and the entropy-augmented one at S/T, different when T0>0 and S>0. Numerically with T1=60, T0=40 the maximisers are 3.318 and 1.991, and the cross-entropy gradient at the current estimate is 0 while that of -H is -0.385. (second round, on the claim as corrected; reports A10 and A11, scripts A14-A17.)"
    }
   ],
   "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": []
  }
 ],
 "definitions": [
  {
   "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": "D4",
   "term": "entropy-augmented likelihood",
   "definition": "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.",
   "kind": "concept",
   "source": "B4"
  },
  {
   "id": "D32",
   "term": "differential entropy",
   "kind": "concept",
   "definition": "For a size with density $f_z$, $$H_z = -\\E\\log f_z = -\\int f_z(z)\\log f_z(z)\\,dz.$$"
  }
 ],
 "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": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C31",
    "C32"
   ]
  },
  {
   "id": "E2",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E2.1",
     "statement": "$$\\nabla_\\theta Q(\\theta\\mid\\theta^{(k)})\\big|_{\\theta=\\theta^{(k)}}=\\nabla\\ell(\\theta^{(k)}),$$ 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"
     ],
     "rule": "Fisher's identity (differentiation under the integral)"
    },
    {
     "id": "E2.2",
     "statement": "A fixed point of EM is a stationary point of $\\ell$",
     "by": [
      "E2.1"
     ],
     "rule": "EM identity"
    },
    {
     "id": "E2.3",
     "statement": "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"
     ],
     "rule": "differentiation"
    },
    {
     "id": "E2.4",
     "statement": "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"
     ],
     "rule": "normal entropy and differentiation"
    },
    {
     "id": "E2.5",
     "statement": "$S/T\\neq S/T_1$ whenever $T_0>0$ and $S>0$, so the stationary points differ",
     "by": [
      "E2.3",
      "E2.4"
     ],
     "rule": "comparison"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C31",
    "C32"
   ]
  },
  {
   "id": "E3",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E3.1",
     "statement": "$H_z=\\frac12\\log(2\\pi e\\sigma^2)$ for the normal distribution",
     "by": [
      "D4"
     ],
     "rule": "definition"
    },
    {
     "id": "E3.2",
     "statement": "$$\\frac{\\partial\\ell_H}{\\partial\\sigma^2}=-\\frac{T_1}{2\\sigma^2}+\\frac{\\sum e_t^2}{2\\sigma^4}-\\frac{T_0}{2\\sigma^2},$$ zero at $\\sigma^2=\\sum e_t^2/T$",
     "by": [
      "E3.1"
     ],
     "rule": "differentiation"
    },
    {
     "id": "E3.3",
     "statement": "EM iterates $\\sigma^2_{(k+1)}=(\\sum e_t^2+T_0\\sigma^2_{(k)})/T$ with fixed point $\\frac{1}{T_1}\\sum e_t^2$",
     "by": [
      "B1"
     ],
     "rule": "fixed point"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   }
  }
 ],
 "artefacts": [
  {
   "id": "A1",
   "kind": "manuscript",
   "uri": "https://github.com/openforecast-org/smooth/blob/8a5fbbb334b0d6192a3c3e8a5773f512e548e891/python/docs/INTERMITTENT_LIKELIHOOD.tex",
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   "description": "LaTeX source of the draft 'Estimating exponential smoothing with unobserved periods: the observed-data likelihood of ADAM and iETS' (October 2026).",
   "licence": "LGPL-2.1-only"
  },
  {
   "id": "A10",
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   "availability": {
    "level": "public"
   },
   "licence": "CC-BY-4.0",
   "description": "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."
  },
  {
   "id": "A11",
   "kind": "proof",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks-2.md",
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   "availability": {
    "level": "public"
   },
   "licence": "CC-BY-4.0",
   "description": "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."
  },
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   "id": "A14",
   "kind": "code",
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   "availability": {
    "level": "public"
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   "licence": "LGPL-2.1-only",
   "description": "Numerical spot checks of the first round of the proof check (Python with numpy); the report A10 says which claim each serves."
  },
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   "id": "A16",
   "kind": "code",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk3.py",
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   "availability": {
    "level": "public"
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   "licence": "LGPL-2.1-only",
   "description": "Numerical spot checks of the proof check (Python with numpy), used in both rounds; the reports A10 and A11 say which claim each serves."
  },
  {
   "id": "A17",
   "kind": "code",
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 ],
 "references": [
  {
   "id": "B1",
   "target": "10.1111/j.2517-6161.1977.tb01600.x",
   "id_type": "doi"
  },
  {
   "id": "B4",
   "target": "10.1016/j.ijpe.2023.109013",
   "id_type": "doi"
  }
 ]
}