{
  "analytic": "synthetic-coverage",
  "kind": "declared_analytical_or_constructed_example",
  "experimental_model_result": false,
  "panels": [
    {
      "kind": "line",
      "x": [
        1.0,
        1.0803315190764675,
        1.1671161911100678,
        1.2608724076806803,
        1.3621602035512723,
        1.4715846019280563,
        1.5897992284504756,
        1.7175102154984982,
        1.8554804201388437,
        2.004533980905239,
        2.1655612406317557,
        2.3395240647448245,
        2.527461586781728,
        2.730496415455323,
        2.949841340341698,
        3.1868065762459103,
        3.4428075895186203,
        3.719373553072642,
        4.018156480603806,
        4.340941094577662,
        4.689655486926549,
        5.066382636136649,
        5.473372849520145,
        5.913057204993991,
        6.388062072657209,
        6.901224802908531,
        7.455610674814367,
        8.054531205964933,
        8.701563933188908,
        9.400573782282974,
        10.155736154404181,
        10.971561867027267,
        11.852924098447012,
        12.805087496773332,
        13.83373962729621,
        14.94502494606524,
        16.145581502618366,
        17.442580591096615,
        18.84376958659311,
        20.357518222611073,
        21.99286858606029,
        23.759589128427628,
        25.668233015746946,
        27.730201165910636,
        29.95781034986427,
        32.364366763473576,
        34.96424550953135,
        37.772976464674564,
        40.80733704412151,
        44.085452418341205,
        47.62690378027989,
        51.45284530985853,
        55.58613053440594,
        60.051448839817596,
        64.87547294786292,
        70.08701824056904,
        75.717214883374,
        81.79969377519475,
        88.37078743614603,
        95.4697470328752,
        103.13897683787222,
        111.42428752325098,
        120.37516980200684,
        130.04509005128978,
        140.4918096835459,
        151.77773017322716,
        163.9702658000207,
        177.14224633510852,
        191.37235207582543,
        206.74558382731308,
        223.35377063851234,
        241.29611832536096,
        260.6798020576923,
        281.6206065495396,
        304.2436176769003,
        328.68396965420544,
        355.0876522326114,
        383.6123827417535,
        414.42854818394187,
        447.720223008213,
        483.6862686437176,
        522.5415213602959,
        564.5180755516963,
        609.8666701068883,
        658.8581861506815,
        711.7852651001318,
        768.9640567018716,
        830.736107491936,
        897.4704009584349,
        969.5655615935915,
        1047.452236000633,
        1131.5956652786067,
        1222.4984640507832,
        1320.7036227366311,
        1426.7977510008586,
        1541.4145817536453,
        1665.238756632532,
        1799.0099155778314,
        1943.5271149298264,
        2099.653600438444,
        2268.321963696039,
        2450.5397127942574,
        2647.395290480231,
        2860.0645757603943,
        3089.8179077880163,
        3338.027673990301,
        3606.17650776123,
        3895.8661446875603,
        4208.826990208893,
        4546.928455862411,
        4912.190125853856,
        5306.793820656116,
        5733.096629695033,
        6193.644990970612,
        6691.189901715636,
        7228.703350949573,
        7809.396072084504,
        8436.736721624851,
        9114.472598521184,
        9846.65202794123,
        10637.64854316314,
        11492.187010036998,
        12415.371850064106,
        13412.717530678983,
        14490.18150486198,
        15654.199796841294,
        16911.72544644807,
        18270.270041765412,
        19737.948588157687,
        21323.527981697593,
        23036.479376536947,
        24887.034759027854,
        26886.24806652944,
        29046.061215980462,
        31379.375436648275,
        33900.12833314499,
        36623.37713903369,
        39565.388658322685,
        42743.73643209655,
        46177.40571069106,
        49886.90685844118,
        53894.397868406064,
        58223.8167188866,
        62901.024362344535,
        67953.95920083763,
        73412.80397070115,
        79310.16603333059,
        85681.27214899482,
        92564.1788971279,
        100000.0
      ],
      "series": [
        {
          "label": "Retention q=1",
          "y": [
            0.0001,
            0.00010803271797098377,
            0.00011671064386386851,
            0.0001260855960934426,
            0.0001362135537015989,
            0.00014715499024849177,
            0.00015897523446914866,
            0.0001717448598362036,
            0.00018554010533980309,
            0.0002004433299795471,
            0.00021654350366159423,
            0.00023393673740789773,
            0.0002527268560151338,
            0.00027302601654951924,
            0.00029495537633174404,
            0.0003186458143551557,
            0.00034423871039165833,
            0.00037188678637523667,
            0.00040175501501434025,
            0.00043402160097349437,
            0.00046887904038344535,
            0.0005065352648900823,
            0.0005472148769375666,
            0.0005911604835029903,
            0.0006386341360610507,
            0.00068991888516034,
            0.0007453204586408273,
            0.0008051690732179,
            0.0008698213899050832,
            0.0009396626245485926,
            0.0010151088256055228,
            0.0010966093322173282,
            0.0011846494266148924,
            0.0012797531959445952,
            0.0013824866197301782,
            0.0014934609003865516,
            0.001613336055482767,
            0.0017428247918158028,
            0.0018826966828080517,
            0.002033782672282628,
            0.00219697992930477,
            0.002373257080506998,
            0.002563659848142037,
            0.002769317124031782,
            0.0029914475116025957,
            0.0032313663703155237,
            0.0034904933990115876,
            0.0037703607969920075,
            0.004072622044033745,
            0.004399061342991461,
            0.004751603771144178,
            0.005132326188990366,
            0.005543468957755553,
            0.005987448519422868,
            0.006466870895592634,
            0.00698454616387786,
            0.007543503972793197,
            0.008147010158130026,
            0.008798584525549588,
            0.009502019865474384,
            0.010261402267201179,
            0.011081132799361518,
            0.011965950623258425,
            0.012920957604020648,
            0.013951644481719691,
            0.01506391866032807,
            0.016264133666360408,
            0.01755912032087946,
            0.01895621965786199,
            0.020463317608234777,
            0.022088881451670272,
            0.023841998016857362,
            0.02573241358472923,
            0.02777057541723829,
            0.02996767479581127,
            0.032335691407576854,
            0.034887438862695234,
            0.03763661106136496,
            0.04059782905294293,
            0.04378668794055612,
            0.0472198032809567,
            0.05091485630940304,
            0.05489063718117241,
            0.05916708526299998,
            0.06376532532734508,
            0.06870769829802365,
            0.07401778496565718,
            0.07972042083409618,
            0.08584169997341667,
            0.09240896544083299,
            0.09945078348838013,
            0.10699689840716764,
            0.11507816446568622,
            0.1237264509894358,
            0.13297451620910142,
            0.14285584508603902,
            0.15340444592248606,
            0.16465460020024494,
            0.17664055979213864,
            0.18939618548888154,
            0.20295452072175596,
            0.21734729448931733,
            0.23260434787481066,
            0.24875297924096082,
            0.26581720429157335,
            0.2838169287857369,
            0.30276703387875525,
            0.32267637694683976,
            0.34354671443121526,
            0.36537155780351216,
            0.38813497927935425,
            0.41181039042683837,
            0.4363593243128071,
            0.46173026020689373,
            0.48785753892159617,
            0.5146604262725951,
            0.5420423913993322,
            0.5698906751012871,
            0.5980762300173879,
            0.6264541182825225,
            0.654864451912293,
            0.6831339551200022,
            0.7110782145280937,
            0.738504661365833,
            0.7652162981223337,
            0.791016140213554,
            0.8157122914055066,
            0.83912351166633,
            0.8610850710476036,
            0.8814546181968751,
            0.9001177341125407,
            0.9169927992872164,
            0.9320347848674015,
            0.9452375950536396,
            0.9566346459734976,
            0.9662974691574127,
            0.9743322732062715,
            0.980874575548341,
            0.9860822096533457,
            0.9901271969953707,
            0.9931871186120577,
            0.995436700199953,
            0.9970403161103365,
            0.9981460132569543,
            0.9988814652538156,
            0.9993520179000729,
            0.9996407216711573,
            0.9998100131407092,
            0.9999045473973935,
            0.9999546227660411
          ]
        },
        {
          "label": "Retention q=0.5",
          "y": [
            4.9999999999999996e-05,
            5.4016467471320516e-05,
            5.835556574710341e-05,
            6.304320922044296e-05,
            6.810739352073735e-05,
            7.357836261796454e-05,
            7.94887893365626e-05,
            8.587397035376622e-05,
            9.277203684320125e-05,
            0.00010022418201732756,
            0.00010827490692248609,
            0.00011697228594842746,
            0.0001263682536309026,
            0.00013651891445185215,
            0.00014748487747766106,
            0.00015933161782263016,
            0.00017212986708308887,
            0.00018595603505833553,
            0.000200892665258803,
            0.00021702892690055346,
            0.00023446114629950977,
            0.00025329338080993136,
            0.00027363803870081825,
            0.0002956165486325443,
            0.0003193600826855528,
            0.0003450103372049543,
            0.0003727203760610345,
            0.0004026555412877866,
            0.0004349944364515423,
            0.00046992998852162584,
            0.0005076705944668329,
            0.0005484413592877663,
            0.0005924854327180552,
            0.0006400654523898043,
            0.0006914651018630253,
            0.0007469907925680971,
            0.000806973479407525,
            0.0008717706205115707,
            0.0009417682924449517,
            0.0010173834730222017,
            0.0010990665048109514,
            0.0011873037533889708,
            0.001282620475475966,
            0.0013855839131886513,
            0.0014968066318712394,
            0.0016169501202369246,
            0.001746728672922837,
            0.0018869135770147182,
            0.002038337625641516,
            0.002201899983376923,
            0.00237857142991707,
            0.002569400010332804,
            0.002775517122122035,
            0.002998144071312513,
            0.0032385991319863867,
            0.0034983051458121845,
            0.00377879770047196,
            0.004081733928254076,
            0.004408901968534665,
            0.004762231140379424,
            0.005143802874043564,
            0.005555862452708129,
            0.006000831618336424,
            0.0064813220980277295,
            0.007000150109642837,
            0.007560351907721323,
            0.00816520043273822,
            0.00881822312847708,
            0.009523220993631816,
            0.010284288934576202,
            0.011105837486421797,
            0.01199261596886175,
            0.012949737141679206,
            0.013982703421963642,
            0.015097434720765343,
            0.016300297950824626,
            0.0175981382487861,
            0.018998311944540467,
            0.02050872129655805,
            0.02213785099474544,
            0.02389480641085243,
            0.025789353550070716,
            0.02783196062539671,
            0.03003384113766934,
            0.03240699829791601,
            0.03496427057360706,
            0.03771937807535435,
            0.04068696942409798,
            0.04388266864937,
            0.04732312156514836,
            0.05102604094934434,
            0.05501024971423348,
            0.059295721096216704,
            0.06390361471223593,
            0.068856307125084,
            0.07417741532899157,
            0.07989181130877435,
            0.08602562553943571,
            0.0926062369780327,
            0.09966224675632623,
            0.10722343241295382,
            0.11532067911093585,
            0.1239858838757031,
            0.1332518284686567,
            0.14315201609306347,
            0.15372046672851872,
            0.16499146552799604,
            0.17699925841438577,
            0.1897776888151736,
            0.2033597694164129,
            0.21777718295134035,
            0.2330597064257095,
            0.249234553892385,
            0.2663256340034928,
            0.28435272018052155,
            0.3033305334497913,
            0.32326774089525573,
            0.3441658763837447,
            0.3660181948108788,
            0.38880847666947277,
            0.4125098062908886,
            0.43708335463333037,
            0.4624772058912675,
            0.48862527627393687,
            0.515446382710359,
            0.5428435284804582,
            0.5707034811505097,
            0.5988967247969503,
            0.6272778722071836,
            0.6556866222203583,
            0.6839493411368718,
            0.7118813336527811,
            0.7392898466359217,
            0.7659778171435843,
            0.7917483338733929,
            0.816409729153182,
            0.8397811583069397,
            0.8616984580832043,
            0.882020010930754,
            0.9006322842393307,
            0.917454671786562,
            0.9324432479784237,
            0.9455930631781091,
            0.9569386677018153,
            0.9665526562565182,
            0.974542171180674,
            0.9810434818837739,
            0.9862149514505745,
            0.9902288846029983,
            0.9932628952197238
          ]
        },
        {
          "label": "Retention q=0.1",
          "y": [
            9.999999999999999e-06,
            1.0803310851517771e-05,
            1.1671152158872982e-05,
            1.2608707630425246e-05,
            1.3621577369449451e-05,
            1.4715811320387511e-05,
            1.5897945401322735e-05,
            1.717504053837072e-05,
            1.8554724834991736e-05,
            2.004523912792893e-05,
            2.1655486201674894e-05,
            2.339508395518634e-05,
            2.527442283813237e-05,
            2.7304727899415542e-05,
            2.9498125818197872e-05,
            3.186771731735866e-05,
            3.442765539138312e-05,
            3.71932298153211e-05,
            4.0180958438866126e-05,
            4.340868581001038e-05,
            4.689568971636711e-05,
            5.0662796279376316e-05,
            5.473250428750028e-05,
            5.912911950947104e-05,
            6.387889978799562e-05,
            6.901021177840374e-05,
            7.455370026591377e-05,
            8.054247106989886e-05,
            8.70122886241921e-05,
            9.400178940954963e-05,
            0.00010155271250839651,
            0.00010971014865345954,
            0.00011852280925143105,
            0.00012804331698103698,
            0.00013832851969248778,
            0.0001494398294729971,
            0.00016144358889165006,
            0.00017441146659724323,
            0.00018842088461569712,
            0.00020355547988025469,
            0.00021990560272895624,
            0.00023756885532096706,
            0.00025665067315741905,
            0.0002772649531448157,
            0.00029953473191113197,
            0.00032359291837798015,
            0.0003495830849082267,
            0.0003776603216888907,
            0.0004079921593758856,
            0.00044075956542206175,
            0.00047615801993515906,
            0.0005143986773698586,
            0.0005557096208504245,
            0.0006003372164499532,
            0.0006485475753215346,
            0.0007006281321884891,
            0.0007568893493580742,
            0.0008176665561287884,
            0.0008833219342186828,
            0.0009542466606543559,
            0.0010308632204308792,
            0.0011136279021853547,
            0.0012030334911248544,
            0.0012996121745166536,
            0.001403938676188968,
            0.0015166336377073137,
            0.001638367265189039,
            0.0017698632621000515,
            0.0019119030698466733,
            0.0020653304395351576,
            0.002231056359924301,
            0.0024100643683452713,
            0.002603416273208551,
            0.002812258318661715,
            0.003037827824003161,
            0.0032814603325940694,
            0.003544597307240541,
            0.003828794411334289,
            0.0041357304174358105,
            0.0044672167874470915,
            0.004825207971037363,
            0.005211812471535852,
            0.005629304731065565,
            0.006080137889231583,
            0.006566957472157838,
            0.0070926160710420155,
            0.007660189071611511,
            0.00827299149784635,
            0.008934596035004468,
            0.009648852298241957,
            0.010419907413847955,
            0.011252227980170344,
            0.01215062347452995,
            0.013120271170611403,
            0.01416674262775113,
            0.015296031808949003,
            0.016514584877999554,
            0.017829331717509632,
            0.01924771919832351,
            0.020777746216532795,
            0.022428000496250853,
            0.024207697134047735,
            0.0261267188336469,
            0.02819565774635959,
            0.03042585879284853,
            0.032829464294127704,
            0.03541945968305407,
            0.038209720000659905,
            0.041215056803083024,
            0.04445126501302738,
            0.047935169142944024,
            0.051684668193674566,
            0.05571877839026263,
            0.06005767275407753,
            0.06472271632537142,
            0.06973649564102902,
            0.07512284083686509,
            0.080906838480963,
            0.08711483295327475,
            0.09377441386674801,
            0.10091438667727834,
            0.10856472325577336,
            0.11675648879921985,
            0.12552174104473027,
            0.13489339732979008,
            0.14490506462548916,
            0.15559082727386347,
            0.16698498680729593,
            0.1791217479450434,
            0.1920348446845152,
            0.20575710037622072,
            0.2203199158439647,
            0.23575268004845124,
            0.25208209856594205,
            0.2693314363473093,
            0.28751967292915664,
            0.3066605705873201,
            0.3267616589566985,
            0.347823143490159,
            0.3698367498818814,
            0.39278452230246685,
            0.4166376000112346,
            0.44135500459350857,
            0.46688247860341997,
            0.4931514255517655,
            0.5200780106000011,
            0.5475624904713348,
            0.5754888492367098,
            0.6037248228305191,
            0.6321223982334278
          ]
        }
      ],
      "xlabel": "Independent generated samples",
      "ylabel": "P(at least one rare retained)",
      "logx": true,
      "logy": false
    }
  ],
  "title": "Rare-event retention depends on both sampling and filtering",
  "assumptions": "Independent samples, rare probability p=0.0001, constant retention q; one observation does not imply learning.",
  "worked": "n=10000时，q=1的至少一次保留率约0.632，q=0.1时约0.095。相关生成或教师缺少该模式会让独立模型过于乐观。",
  "source_ids": [
    "arxiv-2306.11644",
    "arxiv-2212.10560",
    "arxiv-2107.06499"
  ],
  "formula_source": "Article equations and this versioned script",
  "diagram_nodes": [
    "真实任务与覆盖定义",
    "教师采样及来源代际",
    "筛选与分层保留率",
    "家族去重 / 划分隔离",
    "固定预算混合训练",
    "独立真实集及长尾验收"
  ]
}