{
 "bundle_of": "C9",
 "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": "C9",
   "statement": "For [[D12:iETS(M,N,N)]] in the [[D7:conventional multiplicative ETS]] form, the [[D10:normal-drift mixture density]] of the demand size after an interval had a largest [[D14:Kullback-Leibler divergence]] per observation from the exact density at least three orders of magnitude smaller than that of [[D11:the one-step density]], for the Gamma, the log-normal and the inverse Gaussian distribution.",
   "about": {
    "subject": "D10",
    "relation": "has a smaller Kullback-Leibler divergence from the exact density than",
    "object": [
     "D14"
    ],
    "comparator": "D11",
    "conditions": [
     "D12",
     "D7"
    ]
   },
   "type": "descriptive",
   "role": "principal",
   "novelty": "no_prior_found",
   "scope": {
    "population": "iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes",
    "conditions": [
     "scale $s\\in[0.05,1]$",
     "smoothing parameter $\\alpha\\in[0.1,0.6]$",
     "interval length $j\\in[2,20]$",
     "states known at the previous observed period"
    ],
    "excludes": [
     "models with trend or seasonality",
     "ADAM multiplicative ETS"
    ]
   },
   "support": [
    {
     "kind": "simulation",
     "evidence": "E20",
     "artefacts": [
      "A3",
      "A4",
      "A5"
     ],
     "results": [
      "E20.T1.r1",
      "E20.T1.r2",
      "E20.T1.r3"
     ]
    }
   ],
   "assumptions": [
    "C37"
   ],
   "assessments": [
    {
     "by": "author",
     "strength": "moderate",
     "rationale": "Deterministic archived computation with a refinement check; one model, one grid, and values that differ from those first reported in the source draft."
    }
   ],
   "provenance": "author_stated",
   "relations": [
    {
     "type": "extends",
     "reference": "B4",
     "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": "C37",
   "statement": "The [[D9:drift of the states]] of a pure multiplicative model over an interval is approximately normally distributed, with the states known at the previous observed period.",
   "type": "descriptive",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "methodological",
   "check": "tested",
   "provenance": "author_stated",
   "check_evidence": "E20",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  }
 ],
 "definitions": [
  {
   "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": "D9",
   "term": "drift of the states",
   "definition": "The change $d_t=\\log(\\mu_t/\\hat\\mu_t)$ of the logarithm of the location of a pure multiplicative model accumulated over the unobserved periods since the last observed period, relative to the point forecast $\\hat\\mu_t$ with zero errors.",
   "kind": "concept"
  },
  {
   "id": "D10",
   "term": "normal-drift mixture density",
   "definition": "Approximate predictive density of a demand size after $j_t-1$ unobserved periods: the one-step density $f_z(z_t\\mid\\hat\\mu_t e^{d})$ mixed over a normal [[D9:drift of the states]] $d\\sim\\mathcal N(m_t,w_t)$, with $m_t$, $w_t$ computed by 32-node Gauss-Legendre quadrature on the quantile function of $1+\\epsilon$ and the mixture by Gauss-Hermite quadrature.",
   "kind": "method"
  },
  {
   "id": "D11",
   "term": "one-step density",
   "kind": "method",
   "definition": "The density of a demand size after an interval taken as the one-step density $f_z(z_t\\mid\\hat\\mu_t)$ of the model, without the [[D9:drift of the states]] over the unobserved periods."
  },
  {
   "id": "D12",
   "term": "iETS(M,N,N)",
   "kind": "data",
   "definition": "Intermittent demand sizes $$z_t=l_{t-1}(1+\\epsilon_t)$$ with $\\E(1+\\epsilon_t)=1$ and $\\Var(1+\\epsilon_t)=s$, Gamma, log-normal or inverse Gaussian $1+\\epsilon_t$ (the greybox parameterisation of the package) and the level moving at every period, observed or not: $$l_t=l_{t-1}(1+\\epsilon_t)^\\alpha$$ in the [[D6:ADAM multiplicative ETS]] and $$l_t=l_{t-1}(1+\\alpha\\epsilon_t)$$ in the [[D7:conventional multiplicative ETS]]."
  },
  {
   "id": "D13",
   "term": "exact density by convolution",
   "kind": "design",
   "definition": "The exact density of the size after an interval, computed on a grid of step 0.001 in $\\log z$ by FFT convolution with tails cut at 1e-12, compared with an approximation by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20 (kl_mixture.py); a refinement of 164 cells halved the step and cut the tails at 1e-14."
  },
  {
   "id": "D14",
   "term": "Kullback-Leibler divergence",
   "kind": "variable",
   "definition": "The divergence $\\int g\\log(g/h)$ of an approximate density $h$ from the exact density $g$, per observation; over a grid, its largest value."
  }
 ],
 "evidence": [
  {
   "id": "E20",
   "profile": "simulation",
   "description": "Comparison, for iETS(M,N,N) under the ADAM and conventional multiplicative ETS forms, of the normal-drift mixture density (drift moments by 32-node Gauss-Legendre, mixture by 16-node Gauss-Hermite) and of the one-step density against the exact density of the size after an interval, computed on a grid of step 0.001 in log z by FFT convolution with tails cut at 1e-12, by the largest Kullback-Leibler divergence per observation over $s$ from 0.05 to 1 in steps of 0.05, $\\alpha$ from 0.1 to 0.6 in steps of 0.05 and $j$ from 2 to 20; a 164-cell refinement run halved the step and cut tails at 1e-14. The run recorded smooth commit f9d062ee with the claimset scripts not yet committed; the scripts are archived at commit f1de3ea1.",
   "setup": {
    "studied": [
     "D10",
     "D11"
    ],
    "cases": [
     "D12",
     "D6",
     "D7"
    ],
    "procedure": [
     "D13"
    ],
    "lens": [
     "D14"
    ]
   },
   "model": "iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes",
   "outputs": "largest KL per observation, conventional form: mixture 0.00115 (Gamma), 0.00448 (log-normal), 0.00400 (inverse Gaussian), one-step 2.359, 5.947, 47.545; ADAM form: mixture 0.01452, 0 (closed form), 0.00045, one-step 4.584, 18.635, 630.772",
   "artefacts": [
    "A3",
    "A4",
    "A5"
   ],
   "data_origin": "primary",
   "preregistered": false,
   "availability": {
    "level": "public"
   },
   "runs": 1,
   "seeds": [
    20261015
   ],
   "execution": {
    "code_version": "f9d062ee850675acc2160d851a34093d035a474e",
    "notes": "Run with the claimset scripts not yet committed; the scripts are archived at f1de3ea1 and the package code is identical at both commits (user-reported)."
   },
   "tables": [
    {
     "id": "T1",
     "description": "Largest Kullback-Leibler divergence per observation from the exact density of the size after an interval, of the normal-drift mixture density and of the one-step density, for iETS(M,N,N) with Gamma, log-normal and inverse Gaussian sizes in the conventional and the ADAM form",
     "conditions": {
      "model": "iETS(M,N,N)"
     },
     "row_labels": [
      "distribution",
      "ets_form"
     ],
     "columns": [
      "largest KL divergence per observation, normal-drift mixture vs exact density",
      "largest KL divergence per observation, one-step vs exact density"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "dgamma",
        "conventional"
       ],
       "values": [
        0.00114767,
        2.35934
       ]
      },
      {
       "id": "r2",
       "label": [
        "dlnorm",
        "conventional"
       ],
       "values": [
        0.00447805,
        5.94732
       ]
      },
      {
       "id": "r3",
       "label": [
        "dinvgauss",
        "conventional"
       ],
       "values": [
        0.00399623,
        47.5452
       ]
      },
      {
       "id": "r4",
       "label": [
        "dgamma",
        "adam"
       ],
       "values": [
        0.0145197,
        4.58358
       ]
      },
      {
       "id": "r5",
       "label": [
        "dlnorm",
        "adam"
       ],
       "values": [
        9.63871e-07,
        18.6354
       ]
      },
      {
       "id": "r6",
       "label": [
        "dinvgauss",
        "adam"
       ],
       "values": [
        0.000446034,
        630.772
       ]
      }
     ],
     "artefact": "A4"
    }
   ]
  }
 ],
 "artefacts": [
  {
   "id": "A3",
   "kind": "code",
   "uri": "https://github.com/openforecast-org/smooth/blob/f1de3ea10509cace5120af32fa0423a26797dca3/python/docs/claimset/scripts",
   "version": "f1de3ea10509cace5120af32fa0423a26797dca3",
   "licence": "LGPL-2.1-only",
   "availability": {
    "level": "public"
   },
   "description": "Python scripts (kl_mixture.py, scale_recovery.py, interval_bounds.py, lowrank_timing.py with build_lowrank_variants.sh, kl_laplace.py) that produce the numerical results on the observed-data likelihood of ADAM and iETS."
  },
  {
   "id": "A4",
   "kind": "table",
   "uri": "https://github.com/openforecast-org/smooth/blob/f1de3ea10509cace5120af32fa0423a26797dca3/python/docs/claimset/scripts/results/kl_mixture.csv",
   "version": "f1de3ea10509cace5120af32fa0423a26797dca3",
   "licence": "LGPL-2.1-only",
   "availability": {
    "level": "public"
   },
   "description": "Kullback-Leibler divergence per observation of the normal-drift mixture density and of the one-step density from the exact density of the iETS(M,N,N) demand size after an interval, for 25,080 cells of distribution, ETS form, scale, smoothing parameter and interval length."
  },
  {
   "id": "A5",
   "kind": "table",
   "uri": "https://github.com/openforecast-org/smooth/blob/f1de3ea10509cace5120af32fa0423a26797dca3/python/docs/claimset/scripts/results/kl_mixture_refine.csv",
   "version": "f1de3ea10509cace5120af32fa0423a26797dca3",
   "licence": "LGPL-2.1-only",
   "availability": {
    "level": "public"
   },
   "description": "Refinement check of 164 cells of the iETS(M,N,N) Kullback-Leibler grid with half the grid step and a tail cut of 1e-14, with differences from the base run."
  }
 ]
}