# Parameter space [parameter_draws()](../../reference/parameter_draws.md#tidydraws.parameter_draws) is the entry point for parameter-space plots: it pulls posterior draws out of an ArviZ object into a tidy Polars `DataFrame` -- one row per `chain × draw × coordinate`, each variable a column. This example starts with simulated observed data, fits a real PyMC model, and then uses the resulting posterior draws for densities, intervals, contrasts, and cross-parameter plots. # Full PyMC workflow The data are generated from four groups with deliberately different intercepts and slopes, including one negative slope. That separation makes the posterior plots diagnose real group differences rather than noise around one common line. Code ``` python from pathlib import Path import sys import numpy as np import polars as pl import pymc as pm import tidydraws as td import lets_plot as lp import plotnine as p9 for parent in [Path.cwd(), *Path.cwd().parents]: helper_dir = parent / "docs" / "examples" if (helper_dir / "_pymc_workflow.py").exists(): sys.path.insert(0, str(helper_dir)) break from _pymc_workflow import simulate_grouped_regression lp.LetsPlot.setup_html() workflow = simulate_grouped_regression(seed=2026) observed = workflow.observed truth = workflow.truth ``` The observed data and true generating lines show the group separation before fitting. ``` python ( lp.ggplot(observed.sort(["groups", "x"]).to_pandas(), lp.aes("x", "y")) + lp.geom_point(lp.aes(color="groups"), alpha=0.65, size=2.0) + lp.geom_line(lp.aes(y="mu_true", color="groups"), size=1.0) + lp.labs( x="x", y="y", color="group", title="Observed data from known group differences" ) ) ``` Simulated observed data with true group-specific regression lines. Now we build our PyMC model and fit. ``` python coords = { "groups": workflow.group_names, "obs_ind": observed.get_column("obs_ind").to_numpy(), } with pm.Model(coords=coords) as model: x = pm.Data("x", observed.get_column("x").to_numpy(), dims="obs_ind") group_idx = pm.Data( "group_idx", observed.get_column("group_idx").to_numpy().astype("int64"), dims="obs_ind", ) intercept = pm.Normal("intercept", mu=0.0, sigma=2.0, dims="groups") beta = pm.Normal("beta", mu=0.0, sigma=1.5, dims="groups") sigma = pm.HalfNormal("sigma", sigma=1.0) mu = pm.Deterministic( "mu", intercept[group_idx] + beta[group_idx] * x, dims="obs_ind", ) pm.Normal( "y", mu=mu, sigma=sigma, observed=observed.get_column("y").to_numpy(), dims="obs_ind", ) dt = pm.sample( draws=400, tune=400, random_seed=2026, ) ``` Initializing NUTS using jitter+adapt_diag... Multiprocess sampling (2 chains in 2 jobs) NUTS: [intercept, beta, sigma] ``` ``` Sampling 2 chains for 400 tune and 400 draw iterations (800 + 800 draws total) took 1 seconds. We recommend running at least 4 chains for robust computation of convergence diagnostics The rhat statistic is larger than 1.01 for some parameters. This indicates problems during sampling. See https://arxiv.org/abs/1903.08008 for details ``` python dt ``` ![](data:image/svg+xml;base64,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) ``` xr-text-repr-fallback Group: / ├── Group: /posterior │ Dimensions: (chain: 2, draw: 400, groups: 4, obs_ind: 96) │ Coordinates: │ * chain (chain) int64 16B 0 1 │ * draw (draw) int64 3kB 0 1 2 3 4 5 6 7 ... 393 394 395 396 397 398 399 │ * groups (groups) array([0, 1]) draw (draw) int64 0 1 2 3 4 5 ... 395 396 397 398 399 array([ 0, 1, 2, ..., 397, 398, 399], shape=(400,)) groups (groups) \ array(['North', 'South', 'East', 'West'], dtype=' array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,90, 91, 92, 93, 94, 95]) Data variables: (4) intercept (chain, draw, groups) float64 -1.366 0.2531 1.345 ... 1.261 2.061 array([[[-1.36580275, 0.25313461, 1.3450927 , 2.26423163],[-1.13870896, 0.1222468 , 1.28053665, 1.95033886],[-1.35490972, 0.18993479, 1.30592987, 2.28178346],...,[-1.17676678, 0.22781707, 1.06406174, 2.15113783],[-1.31018072, 0.16044135, 1.23691231, 2.09569796],[-1.18674148, 0.17960164, 1.21980986, 2.08509713]],[[-1.13459044, 0.10703741, 1.23265577, 1.98618864],[-1.1662817 , 0.15335124, 1.28974872, 2.17174628],[-1.24873099, 0.11142924, 1.13696449, 2.14346567],...,[-1.2801636 , 0.19729749, 1.1915974 , 2.16851363],[-1.12876508, 0.19523706, 1.26225795, 2.13784672],[-1.24097549, 0.13669628, 1.26132594, 2.06055095]]],shape=(2, 400, 4)) beta (chain, draw, groups) float64 0.3564 0.6474 ... 1.491 -0.5574 array([[[ 0.35644268, 0.64737107, 1.38425817, -0.53866879],[ 0.50588768, 0.84547637, 1.46425933, -0.65205189],[ 0.41203129, 0.77491992, 1.32723215, -0.55565462],...,[ 0.36095693, 0.83657131, 1.4957138 , -0.58600243],[ 0.33523832, 0.67943551, 1.31494732, -0.53630374],[ 0.38896975, 0.89491362, 1.52224988, -0.65286072]],[[ 0.40368041, 0.85275059, 1.32192995, -0.53208658],[ 0.45193714, 0.83765166, 1.35737806, -0.53809366],[ 0.4329819 , 0.82535945, 1.32257014, -0.70082277],...,[ 0.37046336, 0.90079013, 1.37059813, -0.69251148],[ 0.51297514, 0.79981231, 1.57495339, -0.61369912],[ 0.40229355, 0.81876363, 1.49136471, -0.55744606]]],shape=(2, 400, 4)) sigma (chain, draw) float64 0.4225 0.4352 ... 0.4691 0.4101 array([[0.42248927, 0.43523181, 0.4173602 , 0.37833317, 0.50741105,0.41310267, 0.48334963, 0.41464324, 0.50351979, 0.42527828,0.46260952, 0.37525534, 0.50539013, 0.43688636, 0.41450862,0.36713027, 0.47684447, 0.38440117, 0.49758006, 0.38674658,0.49564312, 0.41214862, 0.40031555, 0.44678302, 0.48156028,0.45491772, 0.41624623, 0.40936508, 0.44036818, 0.42892961,0.37399852, 0.39802194, 0.4223284 , 0.4533288 , 0.38982503,0.49790173, 0.42282786, 0.44198354, 0.44198354, 0.42673632,0.40277337, 0.43854844, 0.47453318, 0.37744225, 0.37744225,0.46759635, 0.47406288, 0.35019361, 0.41765174, 0.42599805,0.44947694, 0.48233273, 0.43365136, 0.41726106, 0.42167213,0.40846034, 0.44734114, 0.38389261, 0.42321774, 0.44532016,0.44911735, 0.39580845, 0.42335585, 0.41772105, 0.41772105,0.41194367, 0.42496019, 0.43380345, 0.40419443, 0.46630365,0.41769713, 0.39527712, 0.46177589, 0.50150082, 0.48953321,0.37704606, 0.40716552, 0.4347478 , 0.41559828, 0.40882584,0.43864648, 0.40012808, 0.40164687, 0.40191167, 0.4337899 ,0.42719052, 0.43471954, 0.42702762, 0.4424868 , 0.39199625,0.40154158, 0.41351221, 0.37010189, 0.44996654, 0.40571317,0.45004586, 0.43668528, 0.40589463, 0.40045083, 0.48571877,...0.44018864, 0.44912124, 0.44157689, 0.43399268, 0.42050346,0.40665257, 0.40843773, 0.45431931, 0.45839385, 0.40682888,0.46603767, 0.53268305, 0.36560483, 0.4121309 , 0.43127499,0.42489991, 0.41847033, 0.39838645, 0.45003949, 0.5170921 ,0.49086354, 0.39937954, 0.45481216, 0.43955217, 0.44992783,0.3991828 , 0.42820147, 0.41131288, 0.44297604, 0.42418955,0.40796308, 0.43292602, 0.39474942, 0.41926434, 0.41070194,0.41758172, 0.48453246, 0.48692326, 0.40158922, 0.44740976,0.39972538, 0.37031006, 0.47302326, 0.40878398, 0.46145699,0.41501406, 0.40195071, 0.39399675, 0.49195381, 0.38337976,0.36468894, 0.37486412, 0.37473589, 0.51486209, 0.46970431,0.41903561, 0.44310325, 0.37467451, 0.39258954, 0.4412365 ,0.40090781, 0.42967519, 0.3916201 , 0.40274288, 0.44628236,0.44628236, 0.38005057, 0.39940306, 0.40869521, 0.425478 ,0.43750222, 0.43721217, 0.43662363, 0.47390599, 0.47221677,0.43377236, 0.42213087, 0.5031576 , 0.43982943, 0.42747092,0.42030289, 0.43671577, 0.38487253, 0.41908727, 0.44772287,0.45064508, 0.41077746, 0.43471849, 0.39556141, 0.4327495 ,0.45907498, 0.44636961, 0.405321 , 0.45167495, 0.39728473,0.42468374, 0.39053802, 0.48287273, 0.46909109, 0.41009318]]) mu (chain, draw, obs_ind) float64 -2.101 -2.01 -2.009 ... 1.037 0.855 array([[[-2.10130433, -2.00983809, -2.00878258, ..., 1.42948416,1.27552246, 1.09927753],[-2.18258279, -2.05276769, -2.05126964, ..., 0.9398874 ,0.75351867, 0.54017636],[-2.20511559, -2.09938484, -2.09816472, ..., 1.42071392,1.26189735, 1.08009489],...,[-1.92158328, -1.82895865, -1.82788977, ..., 1.24303984,1.07554929, 0.88381745],[-2.00192817, -1.91590315, -1.91491043, ..., 1.26461549,1.11132976, 0.93585864],[-1.98936103, -1.88954807, -1.88839624, ..., 1.07339226,0.88679234, 0.6731854 ]],[[-1.9675647 , -1.86397686, -1.86278147, ..., 1.16164129,1.00956091, 0.83546959],[-2.09883129, -1.98286036, -1.98152207, ..., 1.33789005,1.18409273, 1.00803597],[-2.1421674 , -2.03106055, -2.02977839, ..., 1.05743647,0.8571281 , 0.62782864],...,[-2.04459614, -1.94953208, -1.94843505, ..., 1.09536401,0.89743116, 0.67085104],[-2.18726353, -2.05562973, -2.05411069, ..., 1.18682858,1.01142179, 0.81062798],[-2.07108803, -1.96785606, -1.96666478, ..., 1.1967053 ,1.0373767 , 0.85498811]]], shape=(2, 400, 96)) Attributes: (9) created_at : 2026-07-14T15:10:15.032658+00:00 creation_library : ArviZ creation_library_version : 1.2.0 creation_library_language : Python inference_library : pymc inference_library_version : 6.0.1 sample_dims : \['chain', 'draw'\] sampling_time : 1.068134069442749 tuning_steps : 400 /sample_stats(29) Dimensions: - chain: 2 - draw: 400 Coordinates: (2) chain (chain) int64 0 1 array([0, 1]) draw (draw) int64 0 1 2 3 4 5 ... 395 396 397 398 399 array([ 0, 1, 2, ..., 397, 398, 399], shape=(400,)) Data variables: (18) energy (chain, draw) float64 75.28 75.22 72.82 ... 71.42 69.1 array([[75.28070555, 75.22443905, 72.81794052, 77.69182372, 82.00300193,84.27792528, 76.35589098, 76.67898034, 73.86362707, 79.3631222 ,78.39773158, 78.33489971, 76.13465377, 71.8102513 , 73.32462989,73.52802192, 72.90231426, 70.92261254, 76.40288797, 76.30970399,75.19048116, 75.67869183, 73.34517672, 76.53860487, 73.97938445,73.44810198, 76.4442985 , 74.68850401, 75.65145761, 76.91385184,69.93289871, 68.18027281, 70.68390026, 73.49968853, 75.27332339,75.9355325 , 80.11778879, 77.74806225, 74.2648302 , 70.67160235,67.47530534, 68.51399286, 67.35899461, 67.91690236, 77.27131558,70.13551271, 72.67080684, 81.76435838, 81.4466749 , 69.38323376,72.63639788, 74.62471012, 77.15977386, 70.92597783, 67.70444794,70.35933469, 68.58742923, 68.21931322, 70.79970786, 71.62043253,73.29361255, 73.64030836, 73.6286315 , 70.56550669, 75.32217164,69.47923551, 68.33187525, 70.85933101, 71.93913984, 74.52048939,77.1901504 , 74.18640453, 74.78780904, 73.23344154, 73.65678328,71.51247862, 70.59945752, 72.325447 , 70.57171945, 79.7400031 ,67.79391387, 69.47816371, 75.02538969, 75.46552247, 72.65965887,75.18672394, 76.33788248, 71.45430627, 71.7234993 , 71.71595652,71.23432776, 71.92152304, 73.86718871, 72.15391693, 79.94002916,71.86029764, 76.41379984, 70.49605157, 73.29608425, 75.97308236,...69.49403886, 75.0426063 , 79.25816826, 77.73902102, 74.7702524 ,68.64721906, 66.69443238, 69.17952337, 72.16712744, 76.22853002,74.40138298, 74.95562022, 74.59638099, 73.89040544, 69.60967672,73.20155078, 70.92909451, 70.24154681, 74.91744232, 80.70243401,79.35985537, 74.8240141 , 72.36694654, 68.15809056, 71.44786521,72.57038698, 72.0606597 , 72.99472761, 72.05897213, 68.18578584,70.14248361, 70.04544618, 71.126237 , 72.33463279, 74.71601728,74.87877597, 82.58888142, 78.2885732 , 73.08972875, 70.54464824,69.88002033, 68.26013882, 71.12424966, 75.57677537, 72.30657792,71.22759663, 70.56765906, 75.90906918, 77.3539448 , 75.82018026,76.09586159, 71.23073702, 70.12779491, 76.51721193, 76.87908705,78.27976466, 69.85825757, 69.47742939, 71.28827842, 70.4172345 ,72.19757513, 67.08262141, 68.50155894, 69.6156483 , 71.66372869,76.1521345 , 70.06696022, 69.5519063 , 68.82728461, 72.14738035,69.86951728, 69.96822049, 77.7628936 , 77.75428798, 77.00057509,78.57803037, 73.98209557, 79.40463092, 78.82387446, 75.56183372,74.54061158, 73.70012015, 71.20667811, 73.25074671, 72.40086646,70.03150271, 72.11450435, 71.99094866, 70.30013319, 67.5455832 ,71.08825752, 77.86538766, 70.06042138, 71.80885394, 73.50068169,68.14236314, 71.32114172, 75.04771371, 71.41863368, 69.10237998]]) reached_max_treedepth (chain, draw) bool False False False ... False False array([[False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,...False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False, False, False, False, False, False,False, False, False, False]]) lp (chain, draw) float64 -70.64 -68.08 ... -69.18 -64.54 array([[-70.64064625, -68.08398082, -67.90918146, -73.48913899,-77.44942482, -71.33573736, -70.11542368, -67.18001729,-71.37231918, -74.03960768, -71.80693032, -68.21445355,-69.81323879, -67.12322941, -67.46454479, -71.57916111,-66.91675699, -66.39596273, -70.68799112, -69.53015785,-70.80436986, -70.331281 , -72.25314417, -71.58873063,-68.84978763, -69.90247562, -70.45462779, -69.70501862,-70.1468975 , -69.02498865, -65.27004988, -65.21267095,-67.87862256, -70.75550292, -70.57048574, -72.3764781 ,-74.86797675, -69.56437464, -69.56437464, -65.77471748,-65.76005427, -65.14251036, -66.4926176 , -65.03988148,-65.03988148, -68.41813517, -68.13016417, -75.90520352,-67.64728842, -67.15870809, -68.41002524, -71.3753578 ,-68.44095854, -66.24692139, -64.83919672, -67.03472807,-66.06239232, -66.27872481, -65.15231077, -70.27452025,-69.1420368 , -68.96491181, -67.0874125 , -66.4979903 ,-66.4979903 , -66.73808319, -66.54703985, -66.24751857,-69.57783193, -70.77986974, -71.45239711, -70.71158524,-70.43521944, -68.86536498, -67.97049748, -67.24940029,-66.37385194, -68.06753307, -69.8292176 , -66.60241418,...-71.56338795, -68.60124663, -65.7141862 , -65.59891855,-69.74434239, -68.58161594, -67.28952172, -70.36708189,-66.41181727, -65.36045979, -66.29594661, -66.5300313 ,-66.60186759, -68.8673515 , -71.12971701, -72.64209766,-74.82352138, -69.08246912, -68.0978443 , -67.87467682,-65.30119683, -66.48169505, -68.90419755, -68.67515157,-66.96171123, 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array([[ 1, -3, 3, 2, -3, 4, 2, 3, 2, -5, 6, -1, -3, 1, 1, 2,2, 4, -3, -4, 2, 5, -1, 3, 2, 2, -2, 7, 4, 5, -2, 5,1, 2, -4, -4, -5, -1, 0, 2, 4, 4, -1, -3, 0, 3, 5, 2,1, -4, -6, -6, 4, -3, 3, 1, -3, 2, 1, 2, 6, -4, -1, 3,0, 1, 4, -4, 2, 4, -2, -1, -4, 6, -1, 3, -2, -4, -1, 1,5, 2, -2, -7, -2, -4, -3, -2, -3, -5, -1, -1, 1, 2, -1, 3,-4, 3, 1, -3, 1, -1, -5, 3, -5, 1, 1, -4, 3, -1, 2, -2,-4, 2, 2, -4, 3, -1, -4, -3, 5, -6, 2, -3, -1, 2, 3, 3,1, -5, -3, 4, 1, 3, -3, -2, 1, 5, -5, -6, 4, 2, -1, 1,-1, 7, 7, -1, -5, 2, -4, -3, 2, 2, -7, -4, 1, -2, 1, 4,-2, -3, 1, 4, 4, 1, -3, -4, 5, 2, -3, -2, -3, -5, -1, 2,3, -1, -3, 5, 5, -2, -1, -2, 3, 1, -1, 3, -1, -6, -1, 4,1, 3, 6, -3, -4, 1, 3, 5, 3, 1, 1, 5, -2, -2, 1, -3,5, -2, 2, 1, 2, 1, -3, 2, -2, 6, -3, 3, 7, -1, -7, -2,5, 0, 1, 7, -4, 4, -1, -1, -3, 3, -3, -6, 1, -6, -6, -2,3, 1, -5, 2, -2, -5, -1, -3, 0, 5, 2, 6, -3, -2, 4, -1,-3, 2, 2, 1, -2, 1, -5, -4, 1, 6, 6, -3, 3, -3, -1, 3,-2, 2, -4, 2, -7, 2, -3, 3, -3, -2, 5, -3, -3, -3, 2, -2,-5, 2, 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-3, -6, -2, -3, -3, -2, -5, 1, 4, 3, -6, 6,6, -4, -6, -5, -2, 4, 1, 4, -5, -3, 3, -5, 3, 0, 5, 2,2, -1, 4, -6, -7, -3, -2, 4, 4, 5, 5, 4, 4, -6, 2, -6,4, 7, -2, -1, 1, -3, 6, -4, 2, 4, -3, -2, 1, 3, -6, 2]]) perf_counter_diff (chain, draw) float64 0.0002303 0.0001826 ... 0.0003309 array([[0.00023031, 0.00018257, 0.00034377, 0.00038803, 0.00035209,0.00039336, 0.00018052, 0.00032524, 0.00034728, 0.00033768,0.0003875 , 0.00033737, 0.00037068, 0.0001809 , 0.00028245,0.00045186, 0.00032454, 0.00037729, 0.00033164, 0.0003948 ,0.00032151, 0.00039373, 0.00018224, 0.00033755, 0.00023152,0.00034219, 0.00018204, 0.0003794 , 0.0003439 , 0.00037847,0.00018301, 0.00033578, 0.00019185, 0.00032859, 0.00039975,0.00033671, 0.00037807, 0.00017789, 0.00033322, 0.00023571,0.00034491, 0.00036775, 0.00020102, 0.00018597, 0.00036947,0.0003328 , 0.00033757, 0.00034407, 0.00018204, 0.00039009,0.0003335 , 0.00035405, 0.00033453, 0.00036046, 0.00032056,0.00030119, 0.0006207 , 0.00065717, 0.00029047, 0.00030427,0.00063113, 0.00062532, 0.00059259, 0.00058345, 0.00029638,0.00054663, 0.00058775, 0.00062808, 0.00060372, 0.0006031 ,0.00061182, 0.00061524, 0.00061223, 0.0006038 , 0.00030208,0.00053162, 0.00058359, 0.00058615, 0.00061246, 0.00054356,0.00058768, 0.00028166, 0.00028593, 0.00061576, 0.00028815,0.00061871, 0.00056921, 0.00028459, 0.00028024, 0.00058394,0.00057774, 0.00028137, 0.00030248, 0.00031232, 0.00031008,0.00055723, 0.00059613, 0.00060316, 0.00025493, 0.00032479,...0.00033108, 0.00035874, 0.00033884, 0.00034276, 0.00037428,0.0003305 , 0.00035201, 0.00017633, 0.00032854, 0.00018688,0.00017935, 0.00032517, 0.00035967, 0.00033224, 0.00023086,0.00033557, 0.00033242, 0.00034717, 0.00033132, 0.00022642,0.00033177, 0.00035722, 0.00017822, 0.0003354 , 0.00022517,0.00033811, 0.00032852, 0.00034125, 0.00017565, 0.00039182,0.00032368, 0.00035016, 0.00018061, 0.00033528, 0.00038588,0.00018043, 0.00032309, 0.00022557, 0.00033292, 0.00037208,0.00018136, 0.00032814, 0.00021574, 0.00055026, 0.00064062,0.00030247, 0.00056453, 0.00063268, 0.00057569, 0.00055237,0.00061406, 0.00062578, 0.00060782, 0.00060528, 0.00057144,0.00058864, 0.00055427, 0.00056137, 0.00056861, 0.00056541,0.00049526, 0.0005859 , 0.0006144 , 0.00054458, 0.00055896,0.00027403, 0.00058046, 0.00029665, 0.00052559, 0.00035472,0.00059329, 0.00053336, 0.00062376, 0.00059598, 0.00031744,0.00063831, 0.00057515, 0.00054601, 0.00053927, 0.00061638,0.00052529, 0.00061426, 0.00032909, 0.00056379, 0.00057682,0.00052587, 0.00034697, 0.00032564, 0.00035731, 0.00056778,0.00062248, 0.00053561, 0.00031917, 0.00056067, 0.00032477,0.00035886, 0.00050454, 0.00055298, 0.00053638, 0.0003309 ]]) smallest_eigval (chain, draw) float64 nan nan nan nan ... nan nan nan nan array([[nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,...nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan]]) step_size_bar (chain, draw) float64 0.7589 0.7589 ... 0.7714 0.7714 array([[0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,0.75887676, 0.75887676, 0.75887676, 0.75887676, 0.75887676,...0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617,0.77138617, 0.77138617, 0.77138617, 0.77138617, 0.77138617]]) n_steps (chain, draw) float64 3.0 3.0 7.0 7.0 ... 7.0 7.0 7.0 3.0 array([[3., 3., 7., 7., 7., 7., 3., 7., 7., 7., 7., 7., 7., 3., 3., 7.,7., 7., 7., 7., 7., 7., 3., 7., 3., 7., 3., 7., 7., 7., 3., 7.,3., 7., 7., 7., 7., 3., 7., 3., 7., 7., 3., 3., 7., 7., 7., 7.,3., 7., 7., 7., 7., 7., 3., 3., 7., 7., 3., 3., 7., 7., 7., 7.,3., 7., 7., 7., 7., 7., 7., 7., 7., 7., 3., 7., 7., 7., 7., 7.,7., 3., 3., 7., 3., 7., 7., 3., 3., 7., 7., 3., 3., 3., 3., 7.,7., 7., 3., 3., 3., 3., 7., 7., 7., 7., 3., 7., 7., 7., 7., 3.,7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 3., 7., 7., 3., 7., 7.,7., 7., 3., 7., 3., 3., 7., 7., 3., 7., 7., 7., 7., 3., 7., 3.,7., 7., 7., 7., 7., 3., 7., 7., 7., 3., 7., 7., 7., 7., 3., 7.,7., 7., 3., 7., 7., 3., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7.,3., 3., 7., 7., 7., 7., 7., 7., 7., 3., 3., 3., 3., 7., 7., 7.,7., 7., 7., 3., 7., 3., 7., 7., 7., 3., 3., 7., 3., 3., 3., 7.,7., 3., 3., 7., 7., 3., 3., 3., 7., 7., 7., 7., 7., 7., 7., 3.,7., 3., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 3., 7., 7., 3.,7., 7., 7., 7., 7., 7., 7., 7., 3., 7., 3., 7., 7., 7., 7., 7.,7., 7., 3., 3., 3., 7., 7., 7., 7., 7., 7., 3., 7., 3., 3., 7.,7., 3., 7., 7., 7., 7., 7., 7., 3., 3., 7., 3., 7., 3., 3., 7.,7., 7., 7., 7., 7., 7., 3., 7., 3., 7., 7., 7., 3., 7., 7., 7.,7., 3., 7., 7., 7., 7., 7., 3., 3., 7., 7., 7., 7., 7., 3., 7.,...7., 7., 3., 3., 3., 7., 7., 7., 3., 3., 3., 7., 7., 3., 3., 7.,7., 3., 3., 7., 3., 3., 7., 7., 7., 3., 7., 7., 7., 7., 7., 7.,7., 7., 3., 3., 3., 3., 7., 7., 7., 7., 7., 7., 7., 7., 3., 7.,7., 7., 3., 7., 7., 3., 3., 7., 7., 3., 7., 3., 3., 7., 7., 7.,7., 3., 7., 7., 3., 7., 7., 7., 3., 3., 7., 3., 7., 7., 7., 3.,3., 7., 7., 3., 3., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7.,7., 3., 7., 3., 7., 7., 7., 7., 7., 7., 3., 7., 3., 7., 3., 7.,7., 7., 7., 7., 3., 3., 7., 3., 7., 7., 7., 7., 7., 3., 3., 7.,3., 7., 3., 7., 3., 3., 7., 7., 7., 3., 3., 3., 7., 7., 7., 3.,3., 3., 7., 7., 3., 7., 7., 3., 7., 7., 7., 7., 7., 7., 7., 3.,3., 3., 3., 3., 3., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7.,7., 7., 3., 3., 7., 7., 3., 7., 3., 7., 7., 3., 7., 7., 7., 7.,7., 7., 3., 7., 7., 7., 7., 3., 7., 7., 3., 3., 7., 7., 7., 7.,7., 7., 3., 7., 7., 3., 3., 7., 7., 7., 7., 3., 7., 7., 7., 7.,7., 7., 7., 3., 7., 3., 3., 7., 7., 7., 3., 7., 7., 7., 7., 3.,7., 7., 3., 7., 3., 7., 7., 7., 3., 7., 7., 7., 3., 7., 7., 3.,7., 3., 7., 7., 3., 7., 3., 7., 7., 3., 7., 7., 7., 7., 7., 7.,7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 7., 3., 7., 3.,7., 3., 7., 7., 7., 7., 3., 7., 7., 7., 7., 7., 7., 7., 3., 7.,7., 7., 3., 3., 3., 7., 7., 7., 3., 7., 3., 3., 7., 7., 7., 3.]]) energy_error (chain, draw) float64 0.4233 -0.2305 ... 0.01767 -0.5857 array([[ 4.23337424e-01, -2.30471359e-01, -7.72487058e-02,1.06189584e+00, 2.26913301e-01, -9.81083730e-01,-3.51655461e-01, -6.91047779e-01, 7.41696288e-01,4.07812529e-01, -3.10325652e-01, -4.17596012e-01,5.47453694e-01, -4.26430670e-01, -1.42349347e-02,8.27814043e-01, -1.04423303e+00, -1.72905654e-01,8.18637478e-01, -3.43751887e-01, 4.45479121e-01,-1.11427060e-01, 5.57963066e-01, -1.31396329e-01,-4.04696454e-01, 8.31055207e-02, 2.53593959e-01,-1.40773341e-01, 1.14497041e-01, -3.32354219e-01,-7.05299523e-01, -1.37812321e-01, 5.94192237e-01,3.38474375e-01, -8.09744898e-02, 1.83481079e-01,4.93222409e-01, -9.02279913e-01, 0.00000000e+00,-3.82692736e-01, -6.39665240e-03, -8.16439269e-02,2.17478730e-01, -2.73940498e-01, 0.00000000e+00,5.14974268e-01, -4.09430606e-02, 1.27853704e+00,-1.30579080e+00, -1.14957241e-01, 2.21366771e-01,4.72899010e-01, -5.77781889e-01, -2.68507374e-01,-2.45052229e-01, 2.65766851e-01, -1.66697770e-01,1.20526549e-01, 2.16622640e-02, 7.03205030e-01,...5.51631908e-01, -1.85136669e-01, -1.93689788e-01,3.34249434e-01, -2.32231506e-01, 3.47800386e-01,4.29811296e-01, -6.98151928e-01, 3.00231824e-01,-8.39879086e-01, 3.61077031e-01, 8.11471725e-01,-1.03221951e-01, -5.46864036e-01, -2.17493010e-01,-3.81299538e-01, 4.61631772e-01, 2.74589447e-02,-3.62354803e-01, 1.77983135e-01, -1.63331424e-02,4.26129620e-02, 1.91802871e-01, 0.00000000e+00,-7.54103582e-02, -2.43480508e-01, 1.20722210e-01,1.59485553e-01, -6.27514126e-02, 1.02343292e-01,2.26110541e-01, 4.75996579e-01, -1.75315421e-02,-2.82134767e-01, 2.17462995e-01, 6.54340558e-01,-7.56169149e-01, 4.30512433e-02, 1.21848815e-01,-6.91790714e-01, 2.53347871e-01, -1.27255702e-01,-8.55553107e-02, -1.19241565e-01, 2.03848632e-01,-4.73931647e-01, -2.70852915e-01, 1.49844657e-01,3.86821641e-01, 7.06236492e-02, -5.23175810e-01,1.61964729e-01, -2.76469898e-01, 1.63320026e-01,4.06841398e-01, 1.82233093e-01, 1.76655558e-02,-5.85704028e-01]]) acceptance_rate (chain, draw) float64 0.5714 0.9075 0.8543 ... 0.9913 1.0 array([[0.57139069, 0.90748251, 0.85426952, 0.65784904, 0.93867676,0.95640916, 0.86959929, 0.76534217, 0.69491961, 0.86367141,1. , 0.85226841, 0.55536614, 0.9743639 , 0.7055426 ,0.77400755, 1. , 0.89738271, 0.37391418, 0.86631772,0.73809659, 0.98886598, 0.85745792, 1. , 1. ,0.92529791, 0.72378139, 1. , 0.9219359 , 0.84414323,1. , 0.86362837, 0.5455676 , 0.84040952, 0.9445474 ,0.92958134, 0.86759057, 1. , 0.77232334, 1. ,0.9983052 , 0.853075 , 0.88822961, 0.99677846, 0.29526946,0.65743867, 0.94403026, 0.40866165, 0.98589246, 1. ,0.80040326, 0.72902655, 0.9185022 , 0.98959369, 0.95897363,0.66454479, 0.99425914, 0.94712627, 0.67925304, 0.5954064 ,0.940835 , 0.94770221, 0.84388895, 0.88760534, 0.42464223,0.92261189, 0.99492547, 0.80508253, 0.69964833, 0.94844937,0.87788564, 0.98287283, 1. , 0.98925372, 0.87997154,0.95332009, 0.84408476, 0.56397852, 0.91484698, 0.57760374,0.99774269, 0.830999 , 0.62926901, 0.83279112, 0.94441492,0.53275322, 0.66410422, 1. , 0.7020028 , 0.91375061,0.74619583, 0.87054444, 1. , 0.37766542, 0.80897347,0.99759442, 0.77409913, 1. , 0.85334305, 0.93906235,...0.95716524, 0.7456295 , 0.97200393, 0.95635385, 0.90319563,0.99217294, 0.96867724, 0.59829881, 0.86394598, 0.71827496,1. , 0.75639576, 1. , 0.66229384, 0.92637055,0.88424708, 0.92837084, 0.98830648, 0.8203719 , 0.55647542,0.96247218, 0.99377226, 0.82139561, 0.94259808, 0.62576393,0.95325089, 0.89310214, 0.80790031, 0.95519059, 0.98015864,0.71761256, 0.84178629, 0.86409509, 0.70853187, 0.78816452,0.92398786, 0.75485535, 0.8843697 , 0.93665887, 0.97922299,0.93547707, 0.84065414, 0.67527547, 0.82297227, 0.93980415,0.81363652, 1. , 0.59031616, 0.8312619 , 0.99401341,0.61999103, 0.99392543, 0.78634473, 0.47592696, 0.96327142,0.87721949, 1. , 0.84317422, 0.6487118 , 0.99164989,0.69406918, 0.89210996, 0.93243745, 0.82940714, 0.72630643,0.48866799, 0.98490713, 0.89674889, 0.81028211, 0.68292088,0.95932738, 0.93254536, 0.57692421, 0.71065823, 0.91266668,0.93452534, 0.96388782, 0.72063877, 0.99577641, 0.96006261,0.98361174, 1. , 0.79434718, 0.89374031, 0.9056318 ,1. , 0.80661838, 0.85487035, 0.7894464 , 0.93072761,0.8070277 , 0.72690715, 1. , 0.63660078, 0.65646142,0.86655368, 0.76607796, 0.74153947, 0.99130429, 1. ]]) tree_depth (chain, draw) int64 2 2 3 3 3 3 2 3 ... 2 3 2 2 3 3 3 2 2, 3, 3, 3, 2, 3,3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3,3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3,2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2,3, 3, 3, 2]]) largest_eigval (chain, draw) float64 nan nan nan nan ... nan nan nan nan array([[nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,...nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,nan, nan, nan, nan, nan, nan, nan, nan, nan, nan]]) perf_counter_start (chain, draw) float64 165.5 165.5 165.5 ... 165.7 165.7 array([[165.45792939, 165.45825115, 165.45852522, 165.45896265,165.45944203, 165.45987863, 165.46036059, 165.46063229,165.4610818 , 165.46151543, 165.46194589, 165.46243411,165.46285713, 165.4633228 , 165.463589 , 165.46400137,165.464551 , 165.4649677 , 165.46544503, 165.46586943,165.46635407, 165.46676675, 165.46725725, 165.46753415,165.46796289, 165.46829067, 165.46872512, 165.46899794,165.46947371, 165.46990667, 165.47038063, 165.47064655,165.47111056, 165.47140335, 165.47181659, 165.47231293,165.4727399 , 165.47320511, 165.47347532, 165.47390067,165.47423189, 165.4746788 , 165.4751498 , 165.47544586,165.47571777, 165.47618391, 165.47660772, 165.47706693,165.47750365, 165.47776996, 165.47825581, 165.47868093,165.47912389, 165.47956559, 165.48065011, 165.48186252,165.4823586 , 165.48680452, 165.48762018, 165.48808515,165.4888574 , 165.48962699, 165.49040123, 165.49116875,165.49187503, 165.49229262, 165.49297081, 165.49369174,165.49446453, 165.49521491, 165.49595888, 165.49671662,165.49747684, 165.49823283, 165.4989766 , 165.49941958,165.50011957, 165.50084551, 165.50157404, 165.50231854,...165.62427336, 165.62469441, 165.6251366 , 165.62543713,165.62585908, 165.626203 , 165.62663122, 165.62708038,165.62750581, 165.62779797, 165.62828556, 165.62869291,165.6291368 , 165.62939926, 165.62981773, 165.63029875,165.63056123, 165.63097266, 165.63128496, 165.63170841,165.63217837, 165.63244817, 165.63286041, 165.6331724 ,165.63384263, 165.63461293, 165.63505284, 165.63573077,165.63649969, 165.63718797, 165.63788576, 165.63863191,165.63937202, 165.64011744, 165.64085237, 165.64155432,165.64228514, 165.64295479, 165.64364841, 165.64435587,165.64505886, 165.64568827, 165.64638649, 165.64716115,165.64781797, 165.64850883, 165.64891103, 165.64963322,165.65009593, 165.65073498, 165.65120257, 165.65191885,165.65257368, 165.6533176 , 165.65404629, 165.65447769,165.6552543 , 165.65594238, 165.65662206, 165.65729862,165.65804902, 165.65869604, 165.65942571, 165.65986659,165.66056215, 165.66125089, 165.6619081 , 165.66238395,165.66284622, 165.66331208, 165.66398904, 165.66474923,165.66540779, 165.6658697 , 165.66654325, 165.66697572,165.66746817, 165.66813241, 165.66879709, 165.66947239]]) process_time_diff (chain, draw) float64 0.0002304 0.0001828 ... 0.0003314 array([[0.0002304 , 0.0001828 , 0.00034414, 0.00038829, 0.00035223,0.00039363, 0.00018054, 0.00032533, 0.00034751, 0.00033777,0.00038832, 0.0003373 , 0.00037108, 0.000181 , 0.00028279,0.00045209, 0.00032466, 0.00037767, 0.00033185, 0.00039491,0.00032165, 0.00039391, 0.00018224, 0.0003377 , 0.00023172,0.00034223, 0.00018225, 0.00037984, 0.00034423, 0.00037877,0.0001831 , 0.00033591, 0.00019186, 0.00032878, 0.00039977,0.00033685, 0.00037864, 0.00017795, 0.00033316, 0.00023599,0.00034524, 0.00036796, 0.00020067, 0.00018602, 0.0003699 ,0.0003329 , 0.00033765, 0.00034411, 0.0001821 , 0.00039032,0.00033369, 0.00035403, 0.00033459, 0.00036064, 0.0003206 ,0.00030112, 0.0006211 , 0.00065733, 0.00029031, 0.00030457,0.00063163, 0.00062582, 0.00059291, 0.0005834 , 0.00029623,0.00054637, 0.0005876 , 0.00062846, 0.00060408, 0.00060313,0.00061208, 0.00061554, 0.00061258, 0.00060402, 0.00030228,0.00053171, 0.00058347, 0.00058651, 0.00061191, 0.00054368,0.00058744, 0.00028188, 0.00028578, 0.0006159 , 0.00028845,0.0006188 , 0.00056941, 0.00028466, 0.00028031, 0.00058394,0.00057809, 0.00028171, 0.00030287, 0.00031256, 0.00031029,0.0005569 , 0.00059627, 0.00060327, 0.0002548 , 0.00032498,...0.00033126, 0.00035883, 0.00033932, 0.00034294, 0.00037463,0.0003306 , 0.00035234, 0.00017638, 0.00032867, 0.00018677,0.00017948, 0.00032537, 0.00035993, 0.0003324 , 0.00023107,0.00033529, 0.00033251, 0.00034722, 0.00033103, 0.00022665,0.00033185, 0.00035733, 0.00017838, 0.0003354 , 0.00022539,0.00033811, 0.00032865, 0.00034126, 0.00017573, 0.00039191,0.00032377, 0.00035026, 0.00018093, 0.00033544, 0.00038617,0.00018072, 0.00032319, 0.00022579, 0.00033309, 0.00037199,0.0001813 , 0.0003282 , 0.00021601, 0.0005509 , 0.00064059,0.0003026 , 0.00056461, 0.00063289, 0.0005758 , 0.00055273,0.00061446, 0.00062598, 0.00060816, 0.00060726, 0.00057192,0.00058881, 0.0005545 , 0.00056178, 0.0005688 , 0.00056553,0.00049542, 0.00058609, 0.00061495, 0.00054484, 0.0005591 ,0.00027439, 0.00058068, 0.00029665, 0.00052558, 0.00035465,0.0005936 , 0.00053374, 0.00062428, 0.00059605, 0.0003175 ,0.0006383 , 0.00057551, 0.00054638, 0.00053932, 0.00061664,0.00052545, 0.00061496, 0.00032939, 0.00056413, 0.00057691,0.00052665, 0.00034718, 0.00032565, 0.00035739, 0.00056807,0.00062293, 0.00053604, 0.00031929, 0.00056104, 0.00032518,0.00035936, 0.00050477, 0.00055311, 0.00053676, 0.00033139]]) Attributes: (9) created_at : 2026-07-14T15:10:15.043039+00:00 creation_library : ArviZ creation_library_version : 1.2.0 creation_library_language : Python inference_library : pymc inference_library_version : 6.0.1 sample_dims : \['chain', 'draw'\] sampling_time : 1.068134069442749 tuning_steps : 400 /observed_data(9) Dimensions: - obs_ind: 96 Coordinates: (1) obs_ind (obs_ind) int64 0 1 2 3 4 5 6 ... 90 91 92 93 94 95 array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,90, 91, 92, 93, 94, 95]) Data variables: (1) y (obs_ind) float64 -2.106 -1.795 ... 1.081 0.7312 array([-2.10603219, -1.79466675, -2.2534471 , -2.09470237, -2.37367922,-1.66374999, -1.26199135, -1.64025567, -0.98177939, -1.33925016,-1.33388682, -1.08498417, -1.05077865, -0.75935677, -1.37400442,-1.85496624, -0.94723062, -1.11395241, -0.86751337, -0.24945719,-1.13196713, -0.38567466, -0.28631984, -0.39290684, -1.11487725,-1.43852721, -1.54137305, -0.6511997 , -0.79023151, -0.31233874,-0.66827013, 0.14492886, -0.40941263, -0.51331568, -0.8449366 ,-0.26285375, 0.47509636, -0.17208811, 0.39568225, 1.46431253,1.12549655, 1.54718495, 0.70766728, 0.49591797, 1.97764339,1.10735944, 1.88247435, 1.87050395, -1.6056923 , -0.43149774,-0.36337803, -0.56331921, -0.69288407, -0.20167962, -0.5726829 ,0.00746766, 0.26016749, 0.29541562, 1.23416368, 0.47333713,1.00227105, 0.60534737, 1.80626905, 2.07709192, 2.32537719,3.60230735, 2.94093613, 2.60894045, 3.9464667 , 3.07664575,3.70327149, 4.03462903, 2.74289723, 3.56837051, 3.03352449,3.0428552 , 2.53761613, 3.15635351, 2.21813466, 1.86553389,2.81637037, 2.17141484, 2.20613488, 2.75030906, 2.38352684,2.1822572 , 2.21455817, 1.46921248, 1.92070899, 1.88258873,1.66678582, 1.34727329, 0.00479285, 1.39274901, 1.08069951,0.73119664]) Attributes: (7) created_at : 2026-07-14T15:10:15.048755+00:00 creation_library : ArviZ creation_library_version : 1.2.0 creation_library_language : Python inference_library : pymc inference_library_version : 6.0.1 sample_dims : \[\] /constant_data(10) Dimensions: - obs_ind: 96 Coordinates: (1) obs_ind (obs_ind) int64 0 1 2 3 4 5 6 ... 90 91 92 93 94 95 array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,90, 91, 92, 93, 94, 95]) Data variables: (2) x (obs_ind) float64 -2.063 -1.807 ... 1.835 2.163 array([-2.0634498 , -1.80684125, -1.80388002, -1.36659913, -1.25328425,-1.15379858, -0.98147769, -0.75830187, -0.63010848, -0.45285532,-0.20326414, -0.04578011, 0.08182629, 0.25403144, 0.44765591,0.55957418, 0.75030867, 1.00038255, 1.11999616, 1.19439373,1.44007857, 1.70470364, 1.80750433, 1.98810138, -1.94865307,-1.68011813, -1.70922901, -1.37040432, -1.40274885, -1.11643657,-1.0500841 , -0.67449204, -0.54198181, -0.34376527, -0.33171222,-0.03219212, 0.04543546, 0.22427874, 0.47530558, 0.67883311,0.79896231, 0.90628277, 1.06436946, 1.41989319, 1.52577654,1.70975215, 2.00076601, 1.93473096, -1.79524338, -1.57401431,-1.52270086, -1.41209235, -1.35745358, -1.05087593, -0.99193711,-0.78434076, -0.63193031, -0.41207621, -0.15782335, -0.13140325,0.00812422, 0.18063304, 0.35732042, 0.49420741, 0.70957397,1.05998299, 1.08296983, 1.32491644, 1.38090935, 1.66574596,1.68682444, 1.94409953, -1.81963689, -1.87272453, -1.5625758 ,-1.44185654, -1.31658669, -1.18260333, -0.85356899, -0.79679937,-0.48650169, -0.49230893, -0.25628245, -0.04971659, 0.11680934,0.16216544, 0.38165759, 0.59301724, 0.71431275, 1.01070775,1.17747632, 1.14778107, 1.33384054, 1.54964885, 1.83546772,2.1626538 ]) group_idx (obs_ind) int32 0 0 0 0 0 0 0 0 ... 3 3 3 3 3 3 3 3 array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,3, 3, 3, 3, 3, 3, 3, 3], dtype=int32) Attributes: (7) created_at : 2026-07-14T15:10:15.050169+00:00 creation_library : ArviZ creation_library_version : 1.2.0 creation_library_language : Python inference_library : pymc inference_library_version : 6.0.1 sample_dims : \[\] # Use tidydraws One [parameter_draws()](../../reference/parameter_draws.md#tidydraws.parameter_draws) call extracts group slopes and intercepts from the fitted posterior: ``` python beta_df = td.parameter_draws(dt, "beta", "intercept") beta_df.head() ``` shape: (5, 5) | chain | draw | groups | beta | intercept | |-------|------|---------|-----------|-----------| | i64 | i64 | str | f64 | f64 | | 0 | 0 | "North" | 0.356443 | -1.365803 | | 0 | 0 | "South" | 0.647371 | 0.253135 | | 0 | 0 | "East" | 1.384258 | 1.345093 | | 0 | 0 | "West" | -0.538669 | 2.264232 | | 0 | 1 | "North" | 0.505888 | -1.138709 | # Plotting ## Posterior density by group The dashed red lines are the true slopes used to simulate the data. - lets-plot - plotnine ``` python ( lp.ggplot(beta_df.to_pandas(), lp.aes("beta", fill="groups")) + lp.geom_density(alpha=0.45) + lp.geom_vline( data=truth.to_pandas(), mapping=lp.aes(xintercept="beta_true"), color="firebrick", linetype="dashed", size=0.8, ) + lp.facet_wrap(facets="groups", ncol=2) + lp.labs(x="beta", y="density", fill="group", title="Posterior slopes by group") ) ``` Posterior density of group-specific slopes, with true values marked. ``` python ( p9.ggplot(beta_df.to_pandas(), p9.aes("beta", fill="groups")) + p9.geom_density(alpha=0.45) + p9.geom_vline( data=truth.to_pandas(), mapping=p9.aes(xintercept="beta_true"), color="firebrick", linetype="dashed", size=0.8, ) + p9.facet_wrap("~groups", ncol=2) + p9.labs(x="beta", y="density", fill="group", title="Posterior slopes by group") ) ```

Posterior density of group-specific slopes, with true values marked.
## Forest plot Summarise each group with one central 89% interval. The red points are the true slopes used to generate the observed data. ``` python beta_forest = td.point_interval(beta_df, "beta", group_by="groups", probs=(0.89,)) ``` - lets-plot - plotnine ``` python ( lp.ggplot(beta_forest.to_pandas(), lp.aes("groups", "beta")) + lp.geom_pointrange( lp.aes(ymin="beta_lower", ymax="beta_upper"), color="steelblue", size=0.9, ) + lp.geom_point( data=truth.to_pandas(), mapping=lp.aes("groups", "beta_true"), color="firebrick", size=2.4, ) + lp.geom_hline(yintercept=0, linetype="dashed", color="#888888") + lp.labs(x="group", y="beta", title="Posterior slope forest plot") ) ``` Posterior slope forest plot with 89% intervals and true values. ``` python ( p9.ggplot(beta_forest.to_pandas(), p9.aes("groups", "beta")) + p9.geom_pointrange( p9.aes(ymin="beta_lower", ymax="beta_upper"), color="steelblue", size=0.9, ) + p9.geom_point( data=truth.to_pandas(), mapping=p9.aes("groups", "beta_true"), color="firebrick", size=2.4, ) + p9.geom_hline(yintercept=0, linetype="dashed", color="#888888") + p9.labs(x="group", y="beta", title="Posterior slope forest plot") ) ```

Posterior slope forest plot with 89% intervals and true values.
## Quantile dotplot Each row below has 100 equally likely dots from the posterior slope distribution for a group. This is a frequency-format alternative to density plots. ``` python quantiles = np.linspace(0.005, 0.995, 100) group_levels = ( beta_df.select("groups").unique().sort("groups").get_column("groups").to_list() ) quantile_dots = pl.DataFrame([ { "groups": group_name, "beta": float( beta_df .filter(pl.col("groups") == group_name) .get_column("beta") .quantile(q) ), "dot": dot, } for group_name in group_levels for dot, q in enumerate(quantiles, start=1) ]) ``` - lets-plot - plotnine ``` python ( lp.ggplot(quantile_dots.to_pandas(), lp.aes("beta", "groups")) + lp.geom_point(size=1.4, alpha=0.65, color="steelblue") + lp.geom_point( data=truth.to_pandas(), mapping=lp.aes("beta_true", "groups"), color="firebrick", size=2.4, ) + lp.labs(x="beta", y="group", title="100-dot posterior summaries") ) ``` Quantile dotplot of group-specific slopes: each dot represents 1% posterior mass. ``` python ( p9.ggplot(quantile_dots.to_pandas(), p9.aes("beta", "groups")) + p9.geom_point(size=1.4, alpha=0.65, color="steelblue") + p9.geom_point( data=truth.to_pandas(), mapping=p9.aes("beta_true", "groups"), color="firebrick", size=2.4, ) + p9.labs(x="beta", y="group", title="100-dot posterior summaries") ) ```

Quantile dotplot of group-specific slopes: each dot represents 1% posterior mass.
## Derived contrasts against a reference group Because every row keeps its `chain` and `draw`, derived quantities are ordinary dataframe operations. Here each contrast is `beta[group] - beta[North]` within draw, summarised with one 89% interval. ``` python reference_group = "North" wide_beta = beta_df.pivot(index=["chain", "draw"], on="groups", values="beta") contrast_draws = pl.concat([ wide_beta.select( "chain", "draw", (pl.col(group_name) - pl.col(reference_group)).alias("contrast"), pl.lit(f"{group_name} - {reference_group}").alias("contrast_name"), ) for group_name in group_levels if group_name != reference_group ]) contrast_forest = td.point_interval( contrast_draws, "contrast", group_by="contrast_name", probs=(0.89,) ) reference_truth = truth.filter(pl.col("groups") == reference_group).get_column( "beta_true" )[0] truth_contrasts = truth.filter(pl.col("groups") != reference_group).select( (pl.col("groups") + " - " + pl.lit(reference_group)).alias("contrast_name"), (pl.col("beta_true") - reference_truth).alias("contrast_true"), ) ``` - lets-plot - plotnine ``` python ( lp.ggplot(contrast_forest.to_pandas(), lp.aes("contrast_name", "contrast")) + lp.geom_pointrange( lp.aes(ymin="contrast_lower", ymax="contrast_upper"), color="steelblue", size=0.9, ) + lp.geom_point( data=truth_contrasts.to_pandas(), mapping=lp.aes("contrast_name", "contrast_true"), color="firebrick", size=2.4, ) + lp.geom_hline(yintercept=0, linetype="dashed", color="#888888") + lp.labs( x="contrast", y="beta difference", title="Posterior slope contrasts against North", ) ) ``` Derived slope contrasts against North, with true contrasts in red. ``` python ( p9.ggplot(contrast_forest.to_pandas(), p9.aes("contrast_name", "contrast")) + p9.geom_pointrange( p9.aes(ymin="contrast_lower", ymax="contrast_upper"), color="steelblue", size=0.9, ) + p9.geom_point( data=truth_contrasts.to_pandas(), mapping=p9.aes("contrast_name", "contrast_true"), color="firebrick", size=2.4, ) + p9.geom_hline(yintercept=0, linetype="dashed", color="#888888") + p9.labs( x="contrast", y="beta difference", title="Posterior slope contrasts against North", ) ) ```

Derived slope contrasts against North, with true contrasts in red.
## Parameters against each other (cross-dim join) Request `beta[groups]` and the scalar `sigma` together: `sigma` is broadcast onto every `beta[groups]` row, so you can colour one by the other directly. ``` python mixed = td.parameter_draws(dt, "beta", "sigma") ``` Cross-join detected between frame 0 and 1. Broadcasting scalar or differently-dimensioned variable on dims ['chain', 'draw']. - lets-plot - plotnine ``` python ( lp.ggplot(mixed.to_pandas(), lp.aes("groups", "beta")) + lp.geom_jitter(lp.aes(color="sigma"), width=0.15, alpha=0.15, size=0.8) + lp.geom_point( data=truth.to_pandas(), mapping=lp.aes("groups", "beta_true"), color="black", size=2.0, ) + lp.scale_color_gradient(low="steelblue", high="firebrick") + lp.labs( x="group", y="beta", color="sigma", title="Slope draws coloured by residual scale", ) ) ``` Posterior slopes by group, coloured by each draw's residual sigma. ``` python ( p9.ggplot(mixed.to_pandas(), p9.aes("groups", "beta")) + p9.geom_jitter(p9.aes(color="sigma"), width=0.15, alpha=0.15, size=0.8) + p9.geom_point( data=truth.to_pandas(), mapping=p9.aes("groups", "beta_true"), color="black", size=2.0, ) + p9.scale_color_gradient(low="steelblue", high="firebrick") + p9.labs( x="group", y="beta", color="sigma", title="Slope draws coloured by residual scale", ) ) ```

Posterior slopes by group, coloured by each draw's residual sigma.
Next, compare fitted posterior draws with model priors using [`compare_draws()`](../../docs/examples/compare_draws.md).