MCMC

from scipy.stats import gamma, poisson
import numpy as np
from empiricaldist import Pmf

alpha = 1.4
dist = gamma(alpha)

lams = np.linspace(0, 10, 101)
prior_pmf = Pmf(dist.pdf(lams), lams)
prior_pmf.normalize()

data = 4
likelihood = poisson.pmf(data, lams)
posterior = prior_pmf * likelihood
posterior.normalize()

0.05015532557804499

sample_prior = dist.rvs(1000)
sample_prior

array([1.78245428, 1.45342136, 1.56617409, 2.26968618, 2.48739122,
       2.04947673, 0.42657469, 1.59840941, 1.61450781, 1.04473805,
       1.38766769, 1.29342865, 0.10011362, 2.84676252, 2.0907023 ,
       0.501516  , 1.75463547, 0.75424778, 2.35723675, 0.20373931,
       1.12665252, 0.3603877 , 0.40185481, 0.14502238, 2.16084403,
       1.33073581, 2.12610331, 0.47025327, 1.47340163, 4.3584236 ,
       3.93169673, 1.61656859, 0.32971242, 0.71269559, 1.80662831,
       1.18335286, 1.46886129, 0.19924373, 1.42013889, 1.16160252,
       3.44516481, 3.50632314, 0.31767409, 0.10326678, 3.80793668,
       1.68673128, 2.42240728, 1.10211366, 0.85346076, 0.60510763,
       0.78364214, 1.01700125, 0.25109979, 1.45602232, 2.53275259,
       0.67428257, 2.13179357, 0.15602002, 2.40751205, 0.00900921,
       1.09614156, 1.1789723 , 4.50074634, 2.28033616, 1.20313575,
       2.03917054, 0.60637777, 1.00250621, 1.08680873, 1.26979198,
       1.60454441, 0.70143311, 0.20483807, 1.38220482, 0.66860397,
       1.97347913, 0.85348215, 1.61166464, 0.5694026 , 0.77472185,
       2.2469975 , 2.59749612, 1.32358443, 1.3492898 , 0.39975522,
       1.77195451, 0.1514805 , 2.74518891, 1.70943013, 1.06480265,
       0.15751771, 2.42403873, 0.93600165, 0.37081925, 4.95480763,
       4.50244729, 0.99897033, 1.75701265, 0.63303305, 0.75107271,
       1.21914169, 0.56481277, 0.20323012, 4.8281205 , 0.05825477,
       0.8180459 , 0.39507497, 0.39567062, 1.70882689, 1.22333205,
       1.25106454, 1.09135668, 0.59421816, 3.10792611, 1.94096089,
       1.33699676, 3.24004141, 0.20338037, 0.77345422, 0.01109566,
       0.28726445, 0.5188063 , 0.32004225, 0.56189764, 0.78222717,
       0.86536462, 0.75354692, 0.36003061, 2.55631833, 5.37122246,
       1.41387676, 1.31543492, 1.01074796, 3.87677072, 0.5995094 ,
       1.20237366, 1.57964719, 0.58273492, 4.03528673, 1.4019504 ,
       0.97618193, 0.15539856, 4.15005826, 1.06660502, 0.7626793 ,
       1.72684506, 0.0450794 , 0.74543249, 0.09050903, 1.34800622,
       2.92492755, 2.42968827, 2.60138038, 2.90331673, 0.89979192,
       1.19012654, 2.53193949, 2.52008528, 1.11941151, 1.85726849,
       2.76141794, 0.46461223, 2.49662685, 2.35624678, 1.86963077,
       2.41364535, 0.30807995, 1.4061337 , 0.27264207, 2.44446813,
       2.24204357, 2.01689348, 1.12378139, 0.69105913, 0.92202871,
       2.93724754, 0.17141724, 0.60518804, 0.53125454, 1.03206292,
       0.47439733, 1.75277398, 0.23028169, 0.8280134 , 1.93138644,
       2.81545321, 1.56380812, 0.18527356, 0.0984316 , 1.99172578,
       0.81991126, 0.4790343 , 1.90666511, 4.16125993, 0.15915635,
       0.25560981, 0.77757414, 1.02994745, 0.7296187 , 2.77953051,
       0.45302316, 2.61951237, 0.82442108, 2.51345799, 0.28933755,
       0.30471831, 0.63519023, 4.19278344, 0.63981378, 1.50199748,
       1.53171353, 0.05193196, 0.23522131, 1.19337516, 0.62119419,
       0.67090562, 4.87535766, 0.12746862, 1.66727365, 0.61737015,
       0.96103356, 0.63758575, 1.58182367, 1.78805179, 0.14303181,
       0.90088986, 2.41355835, 2.88119186, 3.62656613, 0.57027875,
       0.10181002, 2.55424918, 1.17669901, 5.10656878, 0.60856925,
       2.37935349, 0.97901042, 1.04362086, 1.72180322, 0.56624396,
       3.19632108, 0.50092852, 1.61691302, 0.91491038, 1.57061131,
       0.52779649, 1.22280998, 2.5266366 , 0.86564415, 2.48023759,
       4.8845416 , 0.89407679, 1.22596699, 0.75404211, 0.16527154,
       0.76870535, 1.99984975, 0.90019864, 0.13392852, 0.16678515,
       3.25003003, 0.62330084, 3.52010987, 0.60159782, 3.11350541,
       1.44583984, 0.65523014, 2.8895766 , 0.21520675, 0.66540019,
       1.29101126, 3.48560161, 1.58849289, 0.35755348, 0.87234375,
       0.57700548, 0.30313153, 2.42159878, 2.26150262, 0.3824956 ,
       2.97441566, 0.130601  , 1.1687748 , 0.22204499, 0.11365245,
       0.14389429, 1.01204286, 2.34458005, 1.15306329, 0.22943553,
       1.75475892, 0.85484513, 1.95672875, 1.64233556, 1.43204869,
       0.65810987, 1.11893247, 0.42048471, 1.67412056, 1.72722623,
       2.14945311, 0.41395448, 0.74699141, 0.52283731, 2.56181903,
       0.6571291 , 1.08812639, 0.92454495, 1.84451687, 0.5554823 ,
       0.61896596, 0.30390847, 1.06271335, 0.19606563, 1.52173132,
       0.85162107, 1.12267304, 1.35894119, 0.85057098, 0.90606912,
       1.50407866, 0.40416076, 0.63258919, 1.83349409, 3.07069563,
       3.13453365, 3.36162901, 0.91476133, 2.4607268 , 1.37933747,
       1.41308005, 1.40098989, 0.09314009, 1.15655232, 1.35956324,
       0.9562359 , 4.25017262, 1.58582269, 0.88988051, 0.66225242,
       3.1478352 , 0.1912415 , 0.43079422, 1.60481873, 3.48793288,
       3.29679857, 1.42527886, 0.14327117, 1.78002169, 1.60884479,
       1.34999623, 0.41873708, 1.06218268, 3.00648362, 1.35459038,
       0.62904508, 0.68680944, 0.8259365 , 0.43216401, 0.06517087,
       0.76928278, 0.29646467, 0.47003658, 0.49210669, 2.21807655,
       0.8815014 , 1.82491166, 0.38739385, 1.11630756, 2.47541734,
       3.31111865, 3.00380241, 2.0328079 , 0.77396742, 0.62406382,
       1.70915789, 1.16173553, 0.29018906, 2.8465662 , 2.12208868,
       1.23529682, 0.23832301, 0.74280795, 0.42489782, 0.83071628,
       0.10585368, 0.88082506, 1.1422167 , 0.45012877, 0.96804109,
       0.69669037, 1.75769205, 1.79921145, 1.0469775 , 1.27638059,
       1.44456316, 3.18230869, 0.31115153, 2.90534957, 1.09384762,
       0.73737181, 0.73153375, 0.80130282, 0.93084067, 2.82910308,
       2.11995971, 3.48271261, 1.2508142 , 0.77078057, 0.27005484,
       2.30816112, 1.48304143, 2.3197458 , 2.87177339, 0.37235907,
       0.08439045, 1.80168299, 1.27580988, 0.1376812 , 0.72410959,
       1.2551965 , 0.52931964, 1.18787009, 0.42532437, 0.7374523 ,
       0.82102856, 0.82465968, 1.43241383, 1.57478286, 0.98243462,
       1.41672022, 2.12126751, 0.51028047, 0.264361  , 0.17359669,
       3.49280959, 0.28799884, 1.93231242, 0.20307447, 0.67579587,
       0.32516326, 0.66475116, 0.16775069, 1.49053435, 0.48362181,
       1.88346664, 0.7367098 , 1.28901807, 0.41609368, 2.81952939,
       1.92616636, 0.3233559 , 0.43425742, 1.20022375, 1.37352912,
       1.81814333, 0.71787827, 0.20092714, 1.37301219, 1.96938137,
       0.22785241, 1.92048134, 1.11241357, 1.40346287, 2.21565625,
       0.70134262, 0.36980475, 0.88515598, 0.39391247, 2.41861256,
       1.42008427, 0.25944955, 0.98063703, 1.3591506 , 1.06245797,
       2.18967405, 1.14497289, 2.41920052, 1.22700078, 2.14015952,
       3.50280141, 0.32285136, 0.12346727, 0.8337781 , 0.13989125,
       1.66263551, 0.39415498, 2.78120238, 0.24714151, 0.66741164,
       0.32704671, 1.83887574, 4.45999199, 0.7774285 , 0.41141288,
       2.02693585, 0.05295953, 1.66546587, 0.32911593, 0.30873302,
       1.2914885 , 0.14691859, 2.12207068, 0.51315365, 0.80032371,
       2.34913407, 3.4095474 , 0.33763934, 1.22953585, 0.23075278,
       0.7374652 , 0.91715679, 0.39236151, 1.58367854, 0.95839352,
       3.06733169, 0.91288509, 0.19102102, 0.11115259, 3.11374855,
       1.30481605, 0.74220086, 0.77310277, 0.30451101, 1.30518641,
       2.30149429, 0.35855988, 0.89185888, 0.37207391, 1.77390799,
       3.5280598 , 1.96202531, 1.91611586, 0.23112386, 2.80806439,
       0.10056925, 0.11788967, 0.66522683, 2.87382961, 0.68619677,
       2.17313528, 0.9558771 , 1.17700941, 0.94207278, 1.53238011,
       0.99653946, 0.85299825, 0.7120591 , 1.94878342, 0.60649798,
       0.18151201, 0.72210506, 0.90348217, 1.88356921, 0.06351328,
       2.9767626 , 0.59317537, 1.80871685, 0.11252428, 1.13556851,
       0.95026968, 0.58921063, 1.08746956, 0.37980344, 2.12101607,
       1.59854286, 0.12085816, 0.22008041, 1.64744616, 3.37383142,
       2.64912841, 1.68066142, 1.78559911, 3.76130028, 3.00240492,
       0.13001163, 0.04030774, 1.40844236, 0.55993878, 0.19412716,
       0.87202483, 1.25664671, 1.19941488, 7.17456765, 2.52034946,
       1.04923498, 1.0045098 , 1.39849648, 1.06626886, 1.81495584,
       0.22072722, 0.42678823, 0.17427896, 0.31374976, 0.94881118,
       0.11695164, 0.23673133, 0.54695901, 0.5640094 , 1.31881685,
       3.02020657, 0.73039081, 3.26285507, 0.47588791, 1.14561038,
       3.08471669, 1.79722984, 0.71204969, 1.14314614, 0.33993976,
       0.69468141, 1.13031585, 0.20245588, 0.06412572, 0.7679586 ,
       1.08179364, 1.74875858, 0.13766512, 1.02392835, 1.29577827,
       0.09105422, 0.17627404, 1.4013938 , 0.92428193, 1.964099  ,
       0.48575707, 0.22956195, 2.34847801, 1.47081282, 0.05469313,
       6.37212418, 1.58318943, 0.6007103 , 0.24392092, 1.54535754,
       5.88259622, 0.58528123, 0.40431939, 0.20934691, 0.83563943,
       2.18714582, 0.17075649, 0.68424449, 1.29319222, 0.7661801 ,
       0.36229985, 3.38192096, 1.21570705, 0.75593531, 0.72948589,
       0.29429465, 0.24672329, 0.24379611, 5.02651607, 2.94806746,
       2.21708485, 0.4614484 , 1.71470044, 0.51920825, 3.94565173,
       0.708636  , 0.57923338, 2.96705406, 0.2944275 , 1.25637226,
       1.22096943, 0.18290153, 1.77017348, 1.63520181, 1.75808304,
       0.73410077, 5.82731507, 1.52510495, 0.03736303, 0.24150383,
       0.82206031, 0.26913806, 2.28554287, 1.45268443, 0.21564075,
       2.18622272, 0.08782493, 0.54652373, 0.69412775, 1.6556548 ,
       0.27665459, 0.70821905, 1.30150644, 0.53504482, 2.24914298,
       3.48785426, 0.82334135, 0.67226148, 0.54208502, 0.84640797,
       1.11280929, 1.21401206, 1.62551353, 0.20391175, 0.07203695,
       0.04048055, 2.90846273, 0.3421185 , 2.13861689, 5.09107539,
       1.40477632, 1.12705544, 1.71155139, 0.44224997, 0.06196968,
       0.17224229, 0.71676445, 0.86271074, 0.43271951, 1.24696646,
       0.14112179, 2.08775182, 0.79544149, 1.6654211 , 2.12324338,
       2.81877089, 1.25563189, 3.48923833, 0.1365598 , 0.39880259,
       0.20177009, 0.48194506, 1.52881059, 2.32429724, 1.66280629,
       1.83099818, 1.74067556, 3.30083543, 1.8308614 , 1.1145482 ,
       3.32710777, 0.66102754, 1.50616104, 2.36224778, 3.02144457,
       8.86162098, 2.27128422, 1.92403156, 4.45066817, 2.56433866,
       0.21695367, 1.2403261 , 2.52689656, 5.2685631 , 1.84526002,
       0.98541833, 1.15096267, 1.03645475, 0.81933851, 0.3934846 ,
       1.21499941, 1.60832082, 3.18128774, 2.1095381 , 3.8934185 ,
       1.79109006, 1.3959264 , 0.38306127, 1.32617665, 0.08432029,
       2.0966402 , 1.8667662 , 3.3558948 , 0.14799817, 2.14552212,
       1.25160298, 0.91576209, 1.1691564 , 2.29722509, 0.6127987 ,
       4.17148409, 0.85662604, 2.57904496, 2.9415721 , 0.21121228,
       1.09210864, 1.78375426, 2.79602724, 1.05356592, 3.57150381,
       3.26658441, 0.29256895, 1.18776403, 0.87495308, 2.00343635,
       2.06267891, 2.31240083, 5.67768676, 1.90715169, 0.81179953,
       0.86165212, 1.6245984 , 3.67096281, 2.10937595, 1.30683289,
       0.48036824, 3.39090841, 4.63791219, 4.19824051, 2.71398943,
       0.86266062, 0.1049414 , 0.47116751, 1.87292779, 1.40398141,
       3.44842299, 2.04297218, 1.23114444, 0.38680855, 2.4343798 ,
       0.24587911, 1.0012706 , 1.50570632, 2.06932207, 0.68841968,
       0.48475363, 0.2370504 , 1.85823636, 0.99650605, 0.18724216,
       0.31248844, 0.51720466, 1.36847445, 4.38848338, 0.40028451,
       0.57817147, 2.33715831, 0.19275902, 1.14346875, 1.30225192,
       0.78136161, 0.29358472, 1.98502703, 0.51967865, 0.28196284,
       0.88626531, 0.33166157, 2.06054915, 0.25269865, 1.33403699,
       0.0607662 , 0.26593347, 2.59447086, 1.50757733, 0.31869501,
       1.90951547, 1.18127502, 2.99410786, 2.11297465, 3.37042564,
       2.64052509, 1.36346991, 0.68539065, 1.03361491, 0.90540307,
       1.76845903, 2.11128979, 1.78173859, 2.79023275, 0.64396541,
       0.87610746, 5.23750049, 3.41102649, 3.18833522, 1.20921241,
       2.07238565, 1.23863862, 5.66405728, 3.07347514, 0.98414824,
       1.12368196, 1.3060482 , 1.02938722, 1.96038754, 0.81355828,
       1.65056426, 0.11725298, 2.1493593 , 0.49300763, 1.26171529,
       0.27856461, 1.83922204, 0.97770842, 1.27751732, 5.5191054 ,
       2.21460443, 1.72701543, 1.80123051, 1.27982277, 1.69996599,
       0.41897369, 0.56141662, 1.59944028, 1.28269957, 1.09464645,
       0.91348711, 0.83844011, 0.06779634, 1.59391636, 0.92909009,
       0.2249008 , 0.37235769, 3.54665899, 0.69076408, 0.464155  ,
       0.93546219, 1.72142044, 0.65844905, 4.33013673, 0.08389357,
       5.05809403, 0.47798695, 0.63148799, 2.26068668, 0.99370813,
       4.69218164, 2.03889133, 1.06071617, 3.99596801, 2.76045249,
       2.53195504, 6.34568204, 1.79466828, 1.17214677, 0.63195367,
       3.12500801, 2.98122107, 2.74994727, 1.85120285, 0.31523191,
       0.03039504, 2.39494752, 0.05283476, 1.40513227, 0.27310686,
       2.78295955, 2.78503277, 0.17889362, 0.21993918, 2.47193703,
       0.09961994, 0.98290001, 2.87897453, 1.04654061, 1.44689139,
       0.75053196, 1.90072161, 0.07337072, 1.08005232, 1.76350391,
       1.13444569, 2.38286551, 1.20792944, 2.59251206, 3.83880974,
       1.4713979 , 0.82968679, 0.96234638, 0.3518984 , 1.53537545,
       1.53714678, 0.91555435, 0.17847465, 1.04501305, 1.62623627,
       0.56511931, 0.27666544, 1.02573777, 1.76343823, 0.09543004,
       3.27791113, 4.09629567, 1.00900773, 3.87704081, 1.24957286,
       1.9460308 , 0.64492968, 1.87331116, 0.57686789, 0.82118049,
       1.19346087, 0.42413331, 1.62276897, 0.20517721, 0.98301327,
       3.5662663 , 0.61798656, 0.6370218 , 1.46742942, 0.94152712,
       1.78928848, 1.46753359, 1.3924947 , 0.35489384, 9.00032173])