{
 "bundle_of": "C8",
 "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": "C8",
   "statement": "The log-normal scale estimator $2(1-\\sqrt{1-m})$ used in ADAM, with $m=\\frac{1}{T_1}\\sum\\log^2(1+\\epsilon_t)$, is not the maximum likelihood estimator: it overestimates the scale $s$ by about 35% at $s=0.5$ (0.677 against 0.5 when $m=s+s^2/4$) and is undefined for $m>1$.",
   "about": {
    "subject": "the log-normal scale estimator 2(1-sqrt(1-m)) used in ADAM",
    "relation": "is not the maximum likelihood estimator of",
    "object": "the scale s",
    "comparator": null,
    "conditions": []
   },
   "type": "formal",
   "role": "principal",
   "novelty": "refutation",
   "scope": {
    "population": "pure multiplicative ETS in the ADAM form",
    "data_type": "positive univariate time series",
    "conditions": [
     "log-normal distribution with $\\E(1+\\epsilon_t)=1$",
     "every period observed"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E10"
    }
   ],
   "assumptions": [
    "C36"
   ],
   "depends_on": [
    "C15",
    "C7"
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "relations": [
    {
     "type": "contradicts",
     "reference": "B5",
     "locator": "estimator of the scale of the log-normal distribution",
     "self": true,
     "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed."
    }
   ],
   "checks": [
    {
     "by": "P6",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "With f_t=1 and b_t=1/2 the C6 equation is s^2+4s-4m=0, root 2(sqrt(1+m)-1); the ADAM-type estimator is the smaller root of s^2-4s+4m=0, real only for m<=1 and larger than the MLE for every m in (0,1). At m=0.5625 the values are 0.677 and 0.5 (ratio 1.354); a numerical likelihood maximisation on simulated log-normal errors agreed with the formula. (first round; report A10, scripts A14-A16.)"
    }
   ],
   "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": "C6",
   "statement": "For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and [[D2:unobserved period]]s, the [[D8:SSOE Kalman filter]] applied to $\\log z_t$, with the known error mean $-s/2$ added to the prediction and, multiplied by $\\mathbf g$, to the propagation, gives the exact observed-data likelihood, and the maximum likelihood scale $s$ is the positive root of $$\\Big(\\sum \\frac{b_t^2}{f_t}\\Big)s^2+T_1s-\\sum\\frac{(v_t^0)^2}{f_t}=0,$$ where $v_t(s)=v_t^0+sb_t$ is the filter's prediction error.",
   "about": {
    "subject": "D8",
    "relation": "gives in logarithms the exact observed-data likelihood of",
    "object": "D6",
    "comparator": null,
    "conditions": [
     "log-normal distribution",
     "D2"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "no_prior_found",
   "scope": {
    "population": "pure multiplicative ETS in the ADAM form",
    "data_type": "positive univariate time series with unobserved periods",
    "conditions": [
     "log-normal distribution with $\\E(1+\\epsilon_t)=1$"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E9"
    }
   ],
   "assumptions": [
    "C30",
    "C33",
    "C34",
    "C36"
   ],
   "depends_on": [
    "C4",
    "C12"
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "relations": [
    {
     "type": "extends",
     "reference": "B5",
     "self": true,
     "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed."
    }
   ],
   "checks": [
    {
     "by": "P6",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "Derived the filter on log z with mean shift -g s/2 and the Jacobian, and the score in s; the cross terms cancel to give (sum b^2/f)s^2+T1 s-sum v0^2/f=0. Numerically the filter log-likelihood equals the brute-force integral over two unobserved log-sizes (-18.275599457737) and the positive roots (0.29147, 0.25190) equal the grid maxima of the filter likelihood. (first round; report A10, scripts A14-A16.)"
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C7",
   "statement": "For the [[D6:ADAM multiplicative ETS]] with the log-normal distribution and all periods observed, the maximum likelihood estimate of the scale is $\\hat s=2(\\sqrt{1+m}-1)$ with $m=\\frac{1}{T_1}\\sum\\log^2(1+\\epsilon_t)$.",
   "about": {
    "subject": "D6",
    "relation": "has the maximum likelihood scale 2(sqrt(1+m)-1)",
    "object": null,
    "comparator": null,
    "conditions": [
     "log-normal distribution",
     "every period observed"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "no_prior_found",
   "scope": {
    "population": "pure multiplicative ETS in the ADAM form",
    "data_type": "positive univariate time series",
    "conditions": [
     "log-normal distribution with $\\E(1+\\epsilon_t)=1$",
     "every period observed"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E10"
    }
   ],
   "assumptions": [
    "C30",
    "C36"
   ],
   "depends_on": [
    "C6"
   ],
   "verification": "agent_checked",
   "checks": [
    {
     "by": "P5",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "The independent proof checker re-derived the proof of E10 step by step and confirmed it."
    }
   ],
   "provenance": "author_stated",
   "relations": [
    {
     "type": "extends",
     "reference": "B5",
     "self": true,
     "note": "The ADAM monograph defines the multiplicative ETS of ADAM with the log-normal, Gamma and inverse Gaussian distributions and estimates it by likelihood with every period observed."
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C12",
   "statement": "In the [[D6:ADAM multiplicative ETS]], the logarithms of the observation and of the components follow a linear single source of error model with innovation $u_t=\\log(1+\\epsilon_t)$ and the measurement, transition and persistence of the corresponding additive model, whereas in the [[D7:conventional multiplicative ETS]] the innovation of component $k$ is $\\log(1+g_k\\epsilon_t)$, which is not proportional to $u_t$.",
   "about": {
    "subject": "D6",
    "relation": "is a linear SSOE model in logarithms, unlike",
    "object": null,
    "comparator": "D7",
    "conditions": []
   },
   "type": "formal",
   "role": "supporting",
   "novelty": "no_prior_found",
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E8"
    }
   ],
   "verification": "agent_checked",
   "provenance": "author_stated",
   "scope": {
    "population": "pure multiplicative ETS in the ADAM and conventional forms"
   },
   "checks": [
    {
     "by": "P6",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "Taking logs of D6 gives the additive ETS(A,Ad,A) recursion with innovation u_t=log(1+eps); in D7 the innovation is log(1+g eps), and log(1+g eps)/(g log(1+eps)) is not constant for 0<g<1. Simulated ETS(M,Md,M) matched the log-linear recursion to 2e-15 and the ratio took values 0.78 to 1.26. (first round; report A10, scripts A14-A16.)"
    }
   ],
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C15",
   "statement": "When every period is observed, the [[D8:SSOE Kalman filter]] keeps $\\mathbf P_t=\\mathbf 0$, has gain $\\mathbf k_t=\\mathbf g$ and $f_t=1$, and coincides with the single source of error recursion and the likelihood of the one-step errors.",
   "about": {
    "subject": "D8",
    "relation": "coincides with the SSOE recursion when",
    "object": "every period is observed",
    "comparator": null,
    "conditions": []
   },
   "type": "formal",
   "role": "secondary",
   "novelty": "no_prior_found",
   "scope": {
    "population": "pure additive single source of error models",
    "data_type": "univariate time series",
    "conditions": [
     "every period observed"
    ]
   },
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E5"
    },
    {
     "kind": "cited",
     "reference": "B2",
     "self": false
    },
    {
     "kind": "cited",
     "reference": "B3",
     "self": false
    }
   ],
   "depends_on": [
    "C4"
   ],
   "verification": "agent_checked",
   "checks": [
    {
     "by": "P5",
     "outcome": "confirmed",
     "extent": "full",
     "notes": "The independent proof checker re-derived the proof of E5 step by step and confirmed it."
    }
   ],
   "provenance": "author_stated",
   "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": []
  },
  {
   "id": "C36",
   "statement": "In the log-normal ADAM multiplicative ETS, $\\log(1+\\epsilon_t)\\sim\\mathcal N(-s/2,s)$, so that $\\E(1+\\epsilon_t)=1$.",
   "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": "D6",
   "term": "ADAM multiplicative ETS",
   "definition": "Pure multiplicative exponential smoothing in the form of the Augmented Dynamic Adaptive Model, in which each component $k$ is updated by the factor $(1+\\epsilon_t)^{g_k}$; for ETS(M,Md,M) with seasonal period $m$, $$\\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\\phi}s_{t-m}(1+\\epsilon_t) \\\\ l_t &= l_{t-1}b_{t-1}^{\\phi}(1+\\epsilon_t)^{\\alpha} \\\\ b_t &= b_{t-1}^{\\phi}(1+\\epsilon_t)^{\\beta} \\\\ s_t &= s_{t-m}(1+\\epsilon_t)^{\\gamma}. \\end{aligned}$$",
   "kind": "method"
  },
  {
   "id": "D7",
   "term": "conventional multiplicative ETS",
   "definition": "Pure multiplicative exponential smoothing in the form of Hyndman et al. (2008), in which each component $k$ is updated by the factor $1+g_k\\epsilon_t$; for ETS(M,Md,M) with seasonal period $m$, $$\\begin{aligned} z_t &= l_{t-1}b_{t-1}^{\\phi}s_{t-m}(1+\\epsilon_t) \\\\ l_t &= l_{t-1}b_{t-1}^{\\phi}(1+\\alpha\\epsilon_t) \\\\ b_t &= b_{t-1}^{\\phi}(1+\\beta\\epsilon_t) \\\\ s_t &= s_{t-m}(1+\\gamma\\epsilon_t). \\end{aligned}$$",
   "kind": "method"
  },
  {
   "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": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C31",
    "C32"
   ]
  },
  {
   "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": "E5",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E5.1",
     "statement": "With $\\mathbf P_t=\\mathbf 0$, $f_t=1$ and $\\mathbf k_t=\\mathbf g$",
     "by": [
      "D8"
     ],
     "rule": "substitution"
    },
    {
     "id": "E5.2",
     "statement": "$\\mathbf P_{t+1}=\\mathbf g\\mathbf g'-\\mathbf g\\mathbf g'=\\mathbf 0$",
     "by": [
      "E5.1"
     ],
     "rule": "substitution"
    },
    {
     "id": "E5.3",
     "statement": "By induction from $\\mathbf P_1=\\mathbf 0$ the filter is the SSOE recursion",
     "by": [
      "E5.2"
     ],
     "rule": "induction"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   }
  },
  {
   "id": "E8",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E8.1",
     "statement": "$\\log z_t=\\log l_{t-1}+\\phi\\log b_{t-1}+\\log s_{t-m}+u_t$",
     "by": [
      "D6"
     ],
     "rule": "logarithm"
    },
    {
     "id": "E8.2",
     "statement": "$\\log l_t=\\log l_{t-1}+\\phi\\log b_{t-1}+\\alpha u_t$, $\\log b_t=\\phi\\log b_{t-1}+\\beta u_t$ in the ADAM form",
     "by": [
      "D6"
     ],
     "rule": "logarithm"
    },
    {
     "id": "E8.3",
     "statement": "$\\log l_t=\\log l_{t-1}+\\phi\\log b_{t-1}+\\log(1+\\alpha\\epsilon_t)$ in the conventional form",
     "by": [
      "D7"
     ],
     "rule": "logarithm"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   }
  },
  {
   "id": "E9",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E9.1",
     "statement": "$\\log z_t$ follows a linear Gaussian SSOE model with known error mean $-s/2$",
     "by": [
      "C12",
      "C36"
     ],
     "rule": "linearity in logarithms"
    },
    {
     "id": "E9.2",
     "statement": "The filter of the linear Gaussian model with the Jacobian $-\\sum\\log z_t$ gives the exact likelihood",
     "by": [
      "C33",
      "C34",
      "C4",
      "E9.1"
     ],
     "rule": "change of variables"
    },
    {
     "id": "E9.3",
     "statement": "Predictions are linear in $-s/2$ with gain-only coefficients, so $v_t(s)=v_t^0+sb_t$",
     "by": [
      "E9.2"
     ],
     "rule": "linearity"
    },
    {
     "id": "E9.4",
     "statement": "The score in $s$ times $2s^2$ gives $(\\sum b_t^2/f_t)s^2+T_1s-\\sum(v_t^0)^2/f_t=0$",
     "by": [
      "E9.3"
     ],
     "rule": "differentiation"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C30",
    "C33",
    "C34",
    "C36"
   ]
  },
  {
   "id": "E10",
   "profile": "formal",
   "description": "Analytical derivation in the source draft (artefact A1).",
   "foundation": "Kolmogorov probability axioms",
   "steps": [
    {
     "id": "E10.1",
     "statement": "$f_t=1$, $b_t=1/2$ give $s^2+4s-4m=0$",
     "by": [
      "C6",
      "C15"
     ],
     "rule": "substitution"
    },
    {
     "id": "E10.2",
     "statement": "The positive root is $2(\\sqrt{1+m}-1)$; $2(1-\\sqrt{1-m})$ solves $s^2-4s+4m=0$ instead",
     "by": [
      "E10.1"
     ],
     "rule": "quadratic formula"
    },
    {
     "id": "E10.3",
     "statement": "At $s=0.5$, $\\E m=s+s^2/4=0.5625$ and $2(1-\\sqrt{1-0.5625})=0.677$",
     "by": [
      "E10.2",
      "C36"
     ],
     "rule": "arithmetic"
    }
   ],
   "artefacts": [
    "A1"
   ],
   "availability": {
    "level": "public"
   },
   "assumptions": [
    "C36"
   ]
  }
 ],
 "artefacts": [
  {
   "id": "A1",
   "kind": "manuscript",
   "uri": "https://github.com/openforecast-org/smooth/blob/8a5fbbb334b0d6192a3c3e8a5773f512e548e891/python/docs/INTERMITTENT_LIKELIHOOD.tex",
   "version": "sha256:44eb855578262cd1668929bc6659cfcaaa4cd3b3e153541deffd91720908d1e3",
   "availability": {
    "level": "public"
   },
   "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",
   "kind": "proof",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/proof-checks.md",
   "version": "sha256:2ccd9457ef876f55c27b3b0004c25c5a991507f6667c59795e7515cc552fd7f1",
   "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",
   "version": "sha256:1ba26c97d75792bd6d91d375288b585b375e459560b131ad4102b5762e7dbbf7",
   "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."
  },
  {
   "id": "A14",
   "kind": "code",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk1.py",
   "version": "sha256:88de8878db174afda5c9a385d6d46bd0b00e744ce5fa8402814f47aa9d65db8f",
   "availability": {
    "level": "public"
   },
   "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."
  },
  {
   "id": "A16",
   "kind": "code",
   "uri": "https://entrylib.org/p/2026.000003/v2/files/files/proof-checks/chk3.py",
   "version": "sha256:c87a752acc6bbb6816fce17692a752be300da8ab53fa3ddb769164084a8b3833",
   "availability": {
    "level": "public"
   },
   "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."
  }
 ],
 "references": [
  {
   "id": "B2",
   "target": "10.1080/01621459.1997.10473684",
   "id_type": "doi"
  },
  {
   "id": "B3",
   "target": "10.1111/j.2517-6161.1985.tb01355.x",
   "id_type": "doi"
  }
 ]
}