Section 5.2 — Bayesian inference computations¶

This notebook contains the code examples from Section 5.2 Bayesian inference computations from the No Bullshit Guide to Statistics.

See also:

• Chp_04.ipynb
• Chp_05.ipynb
• homework_week02_ivan_savov.ipynb

Notebook setup¶

In [1]:

# load Python modules
import os
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt

# load Python modules import os import numpy as np import pandas as pd import seaborn as sns import matplotlib.pyplot as plt

In [2]:

# Figures setup
plt.clf()  # needed otherwise `sns.set_theme` doesn"t work
from plot_helpers import RCPARAMS
RCPARAMS.update({"figure.figsize": (5, 3)})   # good for screen
# RCPARAMS.update({"figure.figsize": (5, 2)})  # good for print
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc=RCPARAMS,
)

# High-resolution please
%config InlineBackend.figure_format = "retina"

#######################################################

# Figures setup plt.clf() # needed otherwise `sns.set_theme` doesn"t work from plot_helpers import RCPARAMS RCPARAMS.update({"figure.figsize": (5, 3)}) # good for screen # RCPARAMS.update({"figure.figsize": (5, 2)}) # good for print sns.set_theme( context="paper", style="whitegrid", palette="colorblind", rc=RCPARAMS, ) # High-resolution please %config InlineBackend.figure_format = "retina" #######################################################

<Figure size 640x480 with 0 Axes>

Definitions¶

TODO

In [ ]:

Posterior inference using MCMC estimation¶

TODO: point-form definitions

Bayesian inference using Bambi¶

TODO: mention Bambi classes

In [3]:

import bambi as bmb

import bambi as bmb

In [4]:

print(bmb.Prior.__doc__)

print(bmb.Prior.__doc__)

Abstract specification of a term prior

    Parameters
    ----------
    name : str
        Name of prior distribution. Must be the name of a PyMC distribution
        (e.g., `"Normal"`, `"Bernoulli"`, etc.)
    auto_scale: bool
        Whether to adjust the parameters of the prior or use them as passed. Default to `True`.
    kwargs : dict
        Optional keywords specifying the parameters of the named distribution.
    dist : pymc.distributions.distribution.DistributionMeta or callable
        A callable that returns a valid PyMC distribution. The signature must contain `name`,
        `dims`, and `shape`, as well as its own keyworded arguments.
    

In [5]:

print("\n".join(bmb.Model.__doc__.splitlines()[0:27]))

print("\n".join(bmb.Model.__doc__.splitlines()[0:27]))

Specification of model class

    Parameters
    ----------
    formula : str or bambi.formula.Formula
        A model description written using the formula syntax from the `formulae` library.
    data : pandas.DataFrame
        A pandas dataframe containing the data on which the model will be fit, with column
        names matching variables defined in the formula.
    family : str or bambi.families.Family
        A specification of the model family (analogous to the family object in R). Either
        a string, or an instance of class `bambi.families.Family`. If a string is passed, a
        family with the corresponding name must be defined in the defaults loaded at `Model`
        initialization. Valid pre-defined families are `"bernoulli"`, `"beta"`,
        `"binomial"`, `"categorical"`, `"gamma"`, `"gaussian"`, `"negativebinomial"`,
        `"poisson"`, `"t"`, and `"wald"`. Defaults to `"gaussian"`.
    priors : dict
        Optional specification of priors for one or more terms. A dictionary where the keys are
        the names of terms in the model, "common," or "group_specific" and the values are
        instances of class `Prior`. If priors are unset, uses automatic priors inspired by
        the R rstanarm library.
    link : str or Dict[str, str]
        The name of the link function to use. Valid names are `"cloglog"`, `"identity"`,
        `"inverse_squared"`, `"inverse"`, `"log"`, `"logit"`, `"probit"`, and
        `"softmax"`. Not all the link functions can be used with all the families.
        If a dictionary, keys are the names of the target parameters and the values are the names
        of the link functions.

Example 1: estimating the probability of a bias of a coin¶

Let's revisit the biased coin example from Section 5.1, where we want to estimate the probability the coin turns up heads.

Data¶

We have a sample that contains $n=50$ observations (coin tosses) from the coin. The outcome 1 corresponds to heads, while the outcome 0 corresponds to tails. We package the data as the ctoss column of a new data frame ctosses.

In [6]:

ctosses = [1,1,0,0,1,0,1,1,1,1,1,0,1,1,0,0,0,1,1,1,
           1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,0,1,0,
           0,1,0,1,0,1,0,1,1,0]
ctosses = pd.DataFrame({"ctoss":ctosses})
ctosses.head(3)

ctosses = [1,1,0,0,1,0,1,1,1,1,1,0,1,1,0,0,0,1,1,1, 1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,0,1,0, 0,1,0,1,0,1,0,1,1,0] ctosses = pd.DataFrame({"ctoss":ctosses}) ctosses.head(3)

Out[6]:

	ctoss
0	1
1	1
2	0

The proportion of heads outcomes is:

In [7]:

ctosses["ctoss"].mean()

ctosses["ctoss"].mean()

Out[7]:

0.68

Model¶

The model is

$$ C \sim \textrm{Bernoulli}(P), \qquad P \sim \mathcal{U}(0,1). $$

In [8]:

import bambi as bmb
bmb.Prior("Uniform", lower=0, upper=1)

import bambi as bmb bmb.Prior("Uniform", lower=0, upper=1)

Out[8]:

Uniform(lower: 0.0, upper: 1.0)

In [9]:

priors1 = {
    "Intercept": bmb.Prior("Uniform", lower=0, upper=1)
}

mod1 = bmb.Model(formula="ctoss ~ 1",
                 family="bernoulli",
                 link="identity",
                 priors=priors1,
                 data=ctosses)
mod1.set_alias({"Intercept": "P"})
mod1

priors1 = { "Intercept": bmb.Prior("Uniform", lower=0, upper=1) } mod1 = bmb.Model(formula="ctoss ~ 1", family="bernoulli", link="identity", priors=priors1, data=ctosses) mod1.set_alias({"Intercept": "P"}) mod1

Out[9]:

       Formula: ctoss ~ 1
        Family: bernoulli
          Link: p = identity
  Observations: 50
        Priors: 
    target = p
        Common-level effects
            Intercept ~ Uniform(lower: 0.0, upper: 1.0)

In [10]:

mod1.build()
mod1.graph()

mod1.build() mod1.graph()

Out[10]:

In [11]:

# mod1.backend.model

# mod1.backend.model

Fitting the model¶

In [12]:

idata1 = mod1.fit(random_seed=42)

idata1 = mod1.fit(random_seed=42)

Modeling the probability that ctoss==1
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [P]

Output()

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 2 seconds.

Exploring InferenceData objects¶

In [13]:

idata1

idata1

Out[13]:

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 40kB
  Dimensions:  (chain: 4, draw: 1000)
  Coordinates:
    * chain    (chain) int64 32B 0 1 2 3
    * draw     (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999
  Data variables:
      P        (chain, draw) float64 32kB 0.7612 0.5686 0.7762 ... 0.7388 0.6457
  Attributes:
      created_at:                  2025-03-12T12:51:55.667158+00:00
      arviz_version:               0.19.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      sampling_time:               1.81168794631958
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

  • Data variables: (1)
    • P
      (chain, draw)
      float64
      0.7612 0.5686 ... 0.7388 0.6457

      array([[0.76122756, 0.56861221, 0.77616961, ..., 0.70772348, 0.62878475,
              0.70789234],
             [0.68937002, 0.73226925, 0.67623114, ..., 0.60182373, 0.59438488,
              0.6632119 ],
             [0.69674422, 0.67356423, 0.65844755, ..., 0.67517019, 0.67517019,
              0.64238598],
             [0.72020439, 0.75444132, 0.7699058 , ..., 0.70605793, 0.73880719,
              0.64569808]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (8)

    created_at :

    2025-03-12T12:51:55.667158+00:00

    arviz_version :

    0.19.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    sampling_time :

    1.81168794631958

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• sample_stats

  <xarray.Dataset> Size: 496kB
  Dimensions:                (chain: 4, draw: 1000)
  Coordinates:
    * chain                  (chain) int64 32B 0 1 2 3
    * draw                   (draw) int64 8kB 0 1 2 3 4 5 ... 995 996 997 998 999
  Data variables: (12/17)
      acceptance_rate        (chain, draw) float64 32kB 0.7452 0.4529 ... 0.9824
      diverging              (chain, draw) bool 4kB False False ... False False
      energy                 (chain, draw) float64 32kB 33.97 36.99 ... 33.49
      energy_error           (chain, draw) float64 32kB 0.2941 -0.4251 ... -0.2058
      index_in_trajectory    (chain, draw) int64 32kB -1 2 2 3 -3 ... 0 1 -1 1 2
      largest_eigval         (chain, draw) float64 32kB nan nan nan ... nan nan
      ...                     ...
      process_time_diff      (chain, draw) float64 32kB 0.000111 ... 0.000184
      reached_max_treedepth  (chain, draw) bool 4kB False False ... False False
      smallest_eigval        (chain, draw) float64 32kB nan nan nan ... nan nan
      step_size              (chain, draw) float64 32kB 1.702 1.702 ... 1.317
      step_size_bar          (chain, draw) float64 32kB 1.415 1.415 ... 1.327
      tree_depth             (chain, draw) int64 32kB 1 2 2 2 2 2 ... 1 1 1 1 1 2
  Attributes:
      created_at:                  2025-03-12T12:51:55.673444+00:00
      arviz_version:               0.19.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      sampling_time:               1.81168794631958
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

  • Data variables: (17)
    • acceptance_rate
      (chain, draw)
      float64
      0.7452 0.4529 ... 0.8597 0.9824

      array([[0.74516657, 0.45290931, 0.9685891 , ..., 1.        , 0.81275417,
              0.74543394],
             [0.35366975, 0.84103863, 0.9864095 , ..., 0.77299459, 0.97959445,
              1.        ],
             [0.88737108, 0.58326352, 0.98718512, ..., 0.97927302, 0.21914696,
              0.96483155],
             [0.8747444 , 0.81408187, 0.8800982 , ..., 1.        , 0.85965816,
              0.98239012]])

    • diverging
      (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]])

    • energy
      (chain, draw)
      float64
      33.97 36.99 34.41 ... 33.44 33.49

      array([[33.97259792, 36.9913773 , 34.4085046 , ..., 33.35895831,
              33.5887212 , 33.7520542 ],
             [35.10040915, 33.31122128, 33.14971609, ..., 33.42777433,
              33.81221173, 33.2367959 ],
             [33.11697833, 33.87341633, 32.88778339, ..., 33.21889799,
              34.12835248, 32.97094125],
             [33.38605933, 33.78022511, 34.37503189, ..., 33.33355514,
              33.43569456, 33.49374829]])

    • energy_error
      (chain, draw)
      float64
      0.2941 -0.4251 ... 0.1512 -0.2058

      array([[ 0.2941475 , -0.42505719,  0.09897285, ..., -0.21767696,
              -0.01082192,  0.0234105 ],
             [ 0.        ,  0.17311769, -0.19261355, ...,  0.25748323,
               0.0584438 , -0.31266731],
             [ 0.01875866, -0.07102908,  0.0128977 , ..., -0.34242052,
               0.        ,  0.05604317],
             [ 0.11361917,  0.20569434,  0.12772179, ..., -0.17976445,
               0.15122046, -0.20577766]])

    • index_in_trajectory
      (chain, draw)
      int64
      -1 2 2 3 -3 -1 -1 ... -1 0 1 -1 1 2

      array([[-1,  2,  2, ..., -1, -2, -2],
             [ 0, -1, -3, ...,  1,  1, -1],
             [-2,  2, -1, ...,  1,  0, -1],
             [-2, -1,  1, ..., -1,  1,  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]])

    • lp
      (chain, draw)
      float64
      -33.9 -34.05 ... -33.42 -32.95

      array([[-33.89695623, -34.05219472, -34.31518136, ..., -33.01049785,
              -33.08536449, -33.01197245],
             [-32.89479742, -33.30839416, -32.86404386, ..., -33.42730218,
              -33.54794717, -32.87424822],
             [-32.93082919, -32.86289244, -32.88778282, ..., -32.86338325,
              -32.86338325, -32.97092543],
             [-33.14053858, -33.73395094, -34.12957899, ..., -32.99636341,
              -33.41757933, -32.94911324]])

    • max_energy_error
      (chain, draw)
      float64
      0.2941 2.003 ... 0.1512 -0.2058

      array([[ 0.2941475 ,  2.00284402, -0.53497001, ..., -0.21767696,
               0.37660698,  0.52560849],
             [ 1.89329128,  0.17311769, -0.19261355, ...,  0.25748323,
               0.0584438 , -0.31402413],
             [ 0.18464581,  1.13803139,  0.0128977 , ..., -0.34242052,
               1.51801273,  0.05604317],
             [ 0.20780599,  0.20569434,  0.12772179, ..., -0.17976445,
               0.15122046, -0.20577766]])

    • n_steps
      (chain, draw)
      float64
      1.0 3.0 3.0 3.0 ... 1.0 1.0 1.0 3.0

      array([[1., 3., 3., ..., 1., 3., 3.],
             [3., 1., 3., ..., 1., 3., 3.],
             [3., 3., 1., ..., 3., 1., 3.],
             [3., 1., 1., ..., 1., 1., 3.]])

    • perf_counter_diff
      (chain, draw)
      float64
      0.0001378 7.529e-05 ... 0.0001853

      array([[1.37834111e-04, 7.52909109e-05, 3.22791981e-04, ...,
              7.34159257e-05, 2.44708033e-04, 1.81333162e-04],
             [7.60001130e-05, 3.16668302e-05, 7.17910007e-05, ...,
              7.68750906e-05, 1.60875032e-04, 1.49334082e-04],
             [1.78915914e-04, 1.51125016e-04, 6.97919168e-05, ...,
              1.43415993e-04, 7.02079851e-05, 1.41290948e-04],
             [1.80375064e-04, 9.42500774e-05, 8.28751363e-05, ...,
              7.30000902e-05, 7.29158055e-05, 1.85291981e-04]])

    • perf_counter_start
      (chain, draw)
      float64
      1.548e+06 1.548e+06 ... 1.548e+06

      array([[1547567.37198667, 1547567.37218238, 1547567.37246692, ...,
              1547567.7886445 , 1547567.78904771, 1547567.78935521],
             [1547567.40217446, 1547567.40227929, 1547567.40242625, ...,
              1547567.85699087, 1547567.85762037, 1547567.85783754],
             [1547567.41182413, 1547567.41206429, 1547567.41227017, ...,
              1547567.973906  , 1547567.97410187, 1547567.97422325],
             [1547567.5908475 , 1547567.59191479, 1547567.59278812, ...,
              1547568.08235779, 1547568.08254775, 1547568.08275554]])

    • process_time_diff
      (chain, draw)
      float64
      0.000111 7.5e-05 ... 0.000184

      array([[1.11e-04, 7.50e-05, 1.95e-04, ..., 7.40e-05, 1.88e-04, 1.64e-04],
             [7.50e-05, 3.20e-05, 7.10e-05, ..., 7.60e-05, 1.61e-04, 1.50e-04],
             [1.78e-04, 1.52e-04, 6.90e-05, ..., 1.43e-04, 7.00e-05, 1.42e-04],
             [1.80e-04, 9.40e-05, 8.40e-05, ..., 7.20e-05, 7.30e-05, 1.84e-04]])

    • 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]])

    • 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]])

    • step_size
      (chain, draw)
      float64
      1.702 1.702 1.702 ... 1.317 1.317

      array([[1.70234384, 1.70234384, 1.70234384, ..., 1.70234384, 1.70234384,
              1.70234384],
             [1.17465906, 1.17465906, 1.17465906, ..., 1.17465906, 1.17465906,
              1.17465906],
             [1.08848323, 1.08848323, 1.08848323, ..., 1.08848323, 1.08848323,
              1.08848323],
             [1.31729389, 1.31729389, 1.31729389, ..., 1.31729389, 1.31729389,
              1.31729389]])

    • step_size_bar
      (chain, draw)
      float64
      1.415 1.415 1.415 ... 1.327 1.327

      array([[1.41481789, 1.41481789, 1.41481789, ..., 1.41481789, 1.41481789,
              1.41481789],
             [1.33229145, 1.33229145, 1.33229145, ..., 1.33229145, 1.33229145,
              1.33229145],
             [1.50093681, 1.50093681, 1.50093681, ..., 1.50093681, 1.50093681,
              1.50093681],
             [1.32705532, 1.32705532, 1.32705532, ..., 1.32705532, 1.32705532,
              1.32705532]])

    • tree_depth
      (chain, draw)
      int64
      1 2 2 2 2 2 2 1 ... 2 2 1 1 1 1 1 2

      array([[1, 2, 2, ..., 1, 2, 2],
             [2, 1, 2, ..., 1, 2, 2],
             [2, 2, 1, ..., 2, 1, 2],
             [2, 1, 1, ..., 1, 1, 2]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (8)

    created_at :

    2025-03-12T12:51:55.673444+00:00

    arviz_version :

    0.19.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    sampling_time :

    1.81168794631958

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• observed_data

  <xarray.Dataset> Size: 800B
  Dimensions:  (__obs__: 50)
  Coordinates:
    * __obs__  (__obs__) int64 400B 0 1 2 3 4 5 6 7 8 ... 42 43 44 45 46 47 48 49
  Data variables:
      ctoss    (__obs__) int64 400B 1 1 0 0 1 0 1 1 1 1 1 ... 0 1 0 1 0 1 0 1 1 0
  Attributes:
      created_at:                  2025-03-12T12:51:55.675308+00:00
      arviz_version:               0.19.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • __obs__: 50
  • Coordinates: (1)
    • __obs__
      (__obs__)
      int64
      0 1 2 3 4 5 6 ... 44 45 46 47 48 49

      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])

  • Data variables: (1)
    • ctoss
      (__obs__)
      int64
      1 1 0 0 1 0 1 1 ... 0 1 0 1 0 1 1 0

      array([1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1,
             1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1,
             0, 1, 0, 1, 1, 0])

  • Indexes: (1)
    • __obs__
      PandasIndex

      PandasIndex(Index([ 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],
            dtype='int64', name='__obs__'))

  • Attributes: (6)

    created_at :

    2025-03-12T12:51:55.675308+00:00

    arviz_version :

    0.19.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  

In [14]:

type(idata1)

type(idata1)

Out[14]:

arviz.data.inference_data.InferenceData

In [15]:

idata1.groups()

idata1.groups()

Out[15]:

['posterior', 'sample_stats', 'observed_data']

In [16]:

type(idata1["posterior"])

type(idata1["posterior"])

Out[16]:

xarray.core.dataset.Dataset

In [17]:

list(idata1["posterior"].coords)

list(idata1["posterior"].coords)

Out[17]:

['chain', 'draw']

In [18]:

list(idata1["posterior"].data_vars)

list(idata1["posterior"].data_vars)

Out[18]:

['P']

In [19]:

type(idata1["posterior"]["P"])

type(idata1["posterior"]["P"])

Out[19]:

xarray.core.dataarray.DataArray

In [20]:

idata1["posterior"]["P"]

idata1["posterior"]["P"]

Out[20]:

<xarray.DataArray 'P' (chain: 4, draw: 1000)> Size: 32kB
array([[0.76122756, 0.56861221, 0.77616961, ..., 0.70772348, 0.62878475,
        0.70789234],
       [0.68937002, 0.73226925, 0.67623114, ..., 0.60182373, 0.59438488,
        0.6632119 ],
       [0.69674422, 0.67356423, 0.65844755, ..., 0.67517019, 0.67517019,
        0.64238598],
       [0.72020439, 0.75444132, 0.7699058 , ..., 0.70605793, 0.73880719,
        0.64569808]])
Coordinates:
  * chain    (chain) int64 32B 0 1 2 3
  * draw     (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999

xarray.DataArray

'P'

• chain: 4
• draw: 1000

• 0.7612 0.5686 0.7762 0.6757 0.7494 ... 0.7432 0.7061 0.7388 0.6457

  array([[0.76122756, 0.56861221, 0.77616961, ..., 0.70772348, 0.62878475,
          0.70789234],
         [0.68937002, 0.73226925, 0.67623114, ..., 0.60182373, 0.59438488,
          0.6632119 ],
         [0.69674422, 0.67356423, 0.65844755, ..., 0.67517019, 0.67517019,
          0.64238598],
         [0.72020439, 0.75444132, 0.7699058 , ..., 0.70605793, 0.73880719,
          0.64569808]])

• Coordinates: (2)
  • chain
    (chain)
    int64
    0 1 2 3

    array([0, 1, 2, 3])

  • draw
    (draw)
    int64
    0 1 2 3 4 5 ... 995 996 997 998 999

    array([  0,   1,   2, ..., 997, 998, 999])

• Indexes: (2)
  • chain
    PandasIndex

    PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

  • draw
    PandasIndex

    PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
           ...
           990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
          dtype='int64', name='draw', length=1000))

• Attributes: (0)

In [21]:

type(idata1["posterior"]["P"].values)

type(idata1["posterior"]["P"].values)

Out[21]:

numpy.ndarray

In [22]:

idata1["posterior"]["P"].values.round(3)

idata1["posterior"]["P"].values.round(3)

Out[22]:

array([[0.761, 0.569, 0.776, ..., 0.708, 0.629, 0.708],
       [0.689, 0.732, 0.676, ..., 0.602, 0.594, 0.663],
       [0.697, 0.674, 0.658, ..., 0.675, 0.675, 0.642],
       [0.72 , 0.754, 0.77 , ..., 0.706, 0.739, 0.646]])

In [23]:

idata1["posterior"]["P"].values.shape

idata1["posterior"]["P"].values.shape

Out[23]:

(4, 1000)

Extracting the samples from the posterior¶

In [24]:

postP = idata1["posterior"]["P"].values.flatten()
postP

postP = idata1["posterior"]["P"].values.flatten() postP

Out[24]:

array([0.76122756, 0.56861221, 0.77616961, ..., 0.70605793, 0.73880719,
       0.64569808])

Visualize the posterior¶

Histogram¶

In [25]:

ax = sns.histplot(x=postP, stat="density")
ax.set_xlabel("$p$")
ax.set_ylabel("$f_{P|\\mathbf{c}}$");

ax = sns.histplot(x=postP, stat="density") ax.set_xlabel("$p$") ax.set_ylabel("$f_{P|\\mathbf{c}}$");

Kernel density plot¶

In [26]:

ax = sns.kdeplot(x=postP)
ax.set_xlabel("$p$")
ax.set_ylabel("$f_{P|\\mathbf{c}}$");

ax = sns.kdeplot(x=postP) ax.set_xlabel("$p$") ax.set_ylabel("$f_{P|\\mathbf{c}}$");

Summarize the posterior¶

In [27]:

len(postP)

len(postP)

Out[27]:

4000

Posterior mean¶

In [28]:

np.mean(postP)

np.mean(postP)

Out[28]:

0.6747512974497554

Posterior standard deviation¶

In [29]:

np.std(postP)

np.std(postP)

Out[29]:

0.06413160885365309

Posterior median¶

In [30]:

np.median(postP)

np.median(postP)

Out[30]:

0.6780571467155877

Posterior quartiles¶

In [31]:

np.quantile(postP, [0.25, 0.5, 0.75])

np.quantile(postP, [0.25, 0.5, 0.75])

Out[31]:

array([0.63207241, 0.67805715, 0.71976479])

Posterior percentiles¶

In [32]:

np.percentile(postP, [3, 97])

np.percentile(postP, [3, 97])

Out[32]:

array([0.54664801, 0.78923245])

Posterior mode¶

In [33]:

from ministats import mode_from_samples
mode_from_samples(postP)

from ministats import mode_from_samples mode_from_samples(postP)

Out[33]:

0.6833825973512753

Credible interval¶

In [34]:

from ministats import hdi_from_samples
hdi_from_samples(postP, hdi_prob=0.9)

from ministats import hdi_from_samples hdi_from_samples(postP, hdi_prob=0.9)

Out[34]:

[0.5733806152569917, 0.7841482011374427]

Example 2: estimating the IQ score¶

Let's return to the investigation of the smart drug effects on IQ scores.

Data¶

We have collected data from $30$ individuals who took the smart drug, which we have packaged into the iq column of the data frame iqs.

In [35]:

iqs = [ 82.6, 105.5,  96.7,  84.0, 127.2,  98.8,  94.3,
       122.1,  86.8,  86.1, 107.9, 118.9, 116.5, 101.0,
        91.0, 130.0, 155.7, 120.8, 107.9, 117.1, 100.1,
       108.2,  99.8, 103.6, 108.1, 110.3, 101.8, 131.7,
       103.8, 116.4]
iqs = pd.DataFrame({"iq":iqs})
iqs["iq"].mean()

iqs = [ 82.6, 105.5, 96.7, 84.0, 127.2, 98.8, 94.3, 122.1, 86.8, 86.1, 107.9, 118.9, 116.5, 101.0, 91.0, 130.0, 155.7, 120.8, 107.9, 117.1, 100.1, 108.2, 99.8, 103.6, 108.1, 110.3, 101.8, 131.7, 103.8, 116.4] iqs = pd.DataFrame({"iq":iqs}) iqs["iq"].mean()

Out[35]:

107.82333333333334

Model¶

We will use the following Bayesian model:

$$ X \sim \mathcal{N}(M, \sigma=15), \qquad M \sim \mathcal{N}(\mu_M=100,\sigma_M=40). $$

We place a broad prior on the mean parameter $M$, by choosing a large value for the standard deviation $\sigma_M$. We assume the IQ scores come from a population with standard deviation $\sigma = 15$.

In [36]:

import bambi as bmb

priors2 = {
    "Intercept": bmb.Prior("Normal", mu=100, sigma=40),
    "sigma": 15,
}

mod2 = bmb.Model(formula="iq ~ 1",
                 family="gaussian",
                 link="identity",
                 priors=priors2,
                 data=iqs)

mod2.set_alias({"Intercept": "M"})
mod2

import bambi as bmb priors2 = { "Intercept": bmb.Prior("Normal", mu=100, sigma=40), "sigma": 15, } mod2 = bmb.Model(formula="iq ~ 1", family="gaussian", link="identity", priors=priors2, data=iqs) mod2.set_alias({"Intercept": "M"}) mod2

Out[36]:

       Formula: iq ~ 1
        Family: gaussian
          Link: mu = identity
  Observations: 30
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 100.0, sigma: 40.0)
        
        Auxiliary parameters
            sigma ~ 15

In [37]:

mod2.build()
mod2.graph()

mod2.build() mod2.graph()

Out[37]:

Fitting the model¶

This time we'll generate $N=2000$ samples from each chain.

In [38]:

idata2 = mod2.fit(draws=2000, random_seed=42)

idata2 = mod2.fit(draws=2000, random_seed=42)

Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [M]

Output()

Sampling 4 chains for 1_000 tune and 2_000 draw iterations (4_000 + 8_000 draws total) took 3 seconds.

Extract the samples from the posterior¶

In [39]:

postM = idata2["posterior"]["M"].values.flatten()
len(postM)

postM = idata2["posterior"]["M"].values.flatten() len(postM)

Out[39]:

8000

In [40]:

postM

postM

Out[40]:

array([111.84188777, 111.84188777, 110.69558723, ..., 105.38269922,
       106.5084091 , 107.49061702])

Visualize the posterior¶

Histogram¶

In [41]:

ax = sns.histplot(x=postM, stat="density")
ax.set_xlabel("$\\mu$")
ax.set_ylabel("$f_{M|\\mathbf{x}}$");

ax = sns.histplot(x=postM, stat="density") ax.set_xlabel("$\\mu$") ax.set_ylabel("$f_{M|\\mathbf{x}}$");

Kernel density plot¶

In [42]:

ax = sns.kdeplot(x=postM)
ax.set_xlabel("$\\mu$")
ax.set_ylabel("$f_{M|\\mathbf{x}}$");

ax = sns.kdeplot(x=postM) ax.set_xlabel("$\\mu$") ax.set_ylabel("$f_{M|\\mathbf{x}}$");

Summarize the posterior¶

Posterior mean¶

In [43]:

np.mean(postM)

np.mean(postM)

Out[43]:

107.72704847150636

Posterior standard deviation¶

In [44]:

np.std(postM)

np.std(postM)

Out[44]:

2.7150176274941304

Posterior median¶

In [45]:

np.median(postM)

np.median(postM)

Out[45]:

107.71902676335739

Posterior quartiles¶

In [46]:

np.quantile(postM, [0.25, 0.5, 0.75])

np.quantile(postM, [0.25, 0.5, 0.75])

Out[46]:

array([105.87041865, 107.71902676, 109.59653928])

Posterior percentiles¶

In [47]:

np.percentile(postM, [3, 97])

np.percentile(postM, [3, 97])

Out[47]:

array([102.60648664, 112.88591885])

Posterior mode¶

In [48]:

from ministats import mode_from_samples
mode_from_samples(postM)

from ministats import mode_from_samples mode_from_samples(postM)

Out[48]:

107.7582939445229

Credible interval¶

In [49]:

from ministats import hdi_from_samples
hdi_from_samples(postM, hdi_prob=0.9)

from ministats import hdi_from_samples hdi_from_samples(postM, hdi_prob=0.9)

Out[49]:

[103.25275390621668, 112.12840034704796]

Visualizing and interpreting posteriors¶

In [50]:

import arviz as az

import arviz as az

Example 1 continued: inferences about the biased coin¶

Extracting samples (CUTTABLE)¶

For the purpose of our data analysis, we don't care about the chain and draw information and are only interested in in the variable P, which are the actual samples from the posterior. The easiest way to extract the samples from the posterior is to use the ArviZ helper method az.extract shown below.

In [51]:

postP = az.extract(idata1, var_names="P").values
postP.round(3)

postP = az.extract(idata1, var_names="P").values postP.round(3)

Out[51]:

array([0.761, 0.569, 0.776, ..., 0.706, 0.739, 0.646])

The call to the function az.extract selected the posterior group, then selected the variable P within the posterior group.

/CUTTABLE

Summarizing the posterior¶

In [52]:

az.summary(idata1, kind="stats", hdi_prob=0.9)

az.summary(idata1, kind="stats", hdi_prob=0.9)

Out[52]:

	mean	sd	hdi_5%	hdi_95%
P	0.675	0.064	0.573	0.784

Plotting the posterior¶

In [53]:

az.plot_posterior(idata1, var_names="P", hdi_prob=0.9);

az.plot_posterior(idata1, var_names="P", hdi_prob=0.9);

Options:

• If multiple variables, you specify a list to var_names to select only certain variables to plot
• Set the option point_estimate to "mode" or "median"
• Add the option rope to draw region of practical equivalence, e.g., rope=[97,103]

In [54]:

az.plot_forest(idata1, hdi_prob=0.9, figsize=(6,2.2));

az.plot_forest(idata1, hdi_prob=0.9, figsize=(6,2.2));

Example 2 continued: inferences about the population mean¶

Summarizing the posterior¶

In [55]:

az.summary(idata2, kind="stats", hdi_prob=0.9)

az.summary(idata2, kind="stats", hdi_prob=0.9)

Out[55]:

	mean	sd	hdi_5%	hdi_95%
M	107.727	2.715	103.253	112.128

Plotting the posterior¶

In [56]:

az.plot_posterior(idata2, var_names="M", hdi_prob=0.9);

az.plot_posterior(idata2, var_names="M", hdi_prob=0.9);

In [57]:

az.plot_posterior(idata2, var_names="M", hdi_prob=0.9, kind="hist", bins=70);

az.plot_posterior(idata2, var_names="M", hdi_prob=0.9, kind="hist", bins=70);

In [58]:

az.plot_forest(idata2, combined=True, hdi_prob=0.9, figsize=(6,0.6));

az.plot_forest(idata2, combined=True, hdi_prob=0.9, figsize=(6,0.6));

Hypothesis testing¶

In [59]:

az.plot_posterior(idata2, hdi_prob=0.95);

az.plot_posterior(idata2, hdi_prob=0.95);

In [60]:

az.plot_posterior(idata2, hdi_prob="hide", rope=[97,103]);

az.plot_posterior(idata2, hdi_prob="hide", rope=[97,103]);

Explanations¶

Visualizing prior distributions¶

In [61]:

mod2.plot_priors(random_seed=43, color="C1", linestyle="dashed");

mod2.plot_priors(random_seed=43, color="C1", linestyle="dashed");

Sampling: [M]

Bambi default priors¶

In [62]:

iqs["iq"].mean(), iqs["iq"].std(ddof=0), 2.5*iqs["iq"].std(ddof=0)

iqs["iq"].mean(), iqs["iq"].std(ddof=0), 2.5*iqs["iq"].std(ddof=0)

Out[62]:

(107.82333333333334, 15.812119472803834, 39.53029868200959)

In [63]:

mod2d = bmb.Model("iq ~ 1", family="gaussian", data=iqs)
mod2d

mod2d = bmb.Model("iq ~ 1", family="gaussian", data=iqs) mod2d

Out[63]:

       Formula: iq ~ 1
        Family: gaussian
          Link: mu = identity
  Observations: 30
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: 107.8233, sigma: 39.5303)
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 15.8121)

In [64]:

mod2d.build()
mod2d.plot_priors();

mod2d.build() mod2d.plot_priors();

Sampling: [Intercept, sigma]

More about inference data objects¶

In [65]:

idata1

idata1

Out[65]:

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 40kB
  Dimensions:  (chain: 4, draw: 1000)
  Coordinates:
    * chain    (chain) int64 32B 0 1 2 3
    * draw     (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999
  Data variables:
      P        (chain, draw) float64 32kB 0.7612 0.5686 0.7762 ... 0.7388 0.6457
  Attributes:
      created_at:                  2025-03-12T12:51:55.667158+00:00
      arviz_version:               0.19.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      sampling_time:               1.81168794631958
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

  • Data variables: (1)
    • P
      (chain, draw)
      float64
      0.7612 0.5686 ... 0.7388 0.6457

      array([[0.76122756, 0.56861221, 0.77616961, ..., 0.70772348, 0.62878475,
              0.70789234],
             [0.68937002, 0.73226925, 0.67623114, ..., 0.60182373, 0.59438488,
              0.6632119 ],
             [0.69674422, 0.67356423, 0.65844755, ..., 0.67517019, 0.67517019,
              0.64238598],
             [0.72020439, 0.75444132, 0.7699058 , ..., 0.70605793, 0.73880719,
              0.64569808]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (8)

    created_at :

    2025-03-12T12:51:55.667158+00:00

    arviz_version :

    0.19.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    sampling_time :

    1.81168794631958

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• sample_stats

  <xarray.Dataset> Size: 496kB
  Dimensions:                (chain: 4, draw: 1000)
  Coordinates:
    * chain                  (chain) int64 32B 0 1 2 3
    * draw                   (draw) int64 8kB 0 1 2 3 4 5 ... 995 996 997 998 999
  Data variables: (12/17)
      acceptance_rate        (chain, draw) float64 32kB 0.7452 0.4529 ... 0.9824
      diverging              (chain, draw) bool 4kB False False ... False False
      energy                 (chain, draw) float64 32kB 33.97 36.99 ... 33.49
      energy_error           (chain, draw) float64 32kB 0.2941 -0.4251 ... -0.2058
      index_in_trajectory    (chain, draw) int64 32kB -1 2 2 3 -3 ... 0 1 -1 1 2
      largest_eigval         (chain, draw) float64 32kB nan nan nan ... nan nan
      ...                     ...
      process_time_diff      (chain, draw) float64 32kB 0.000111 ... 0.000184
      reached_max_treedepth  (chain, draw) bool 4kB False False ... False False
      smallest_eigval        (chain, draw) float64 32kB nan nan nan ... nan nan
      step_size              (chain, draw) float64 32kB 1.702 1.702 ... 1.317
      step_size_bar          (chain, draw) float64 32kB 1.415 1.415 ... 1.327
      tree_depth             (chain, draw) int64 32kB 1 2 2 2 2 2 ... 1 1 1 1 1 2
  Attributes:
      created_at:                  2025-03-12T12:51:55.673444+00:00
      arviz_version:               0.19.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      sampling_time:               1.81168794631958
      tuning_steps:                1000
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 1000
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 995 996 997 998 999

      array([  0,   1,   2, ..., 997, 998, 999])

  • Data variables: (17)
    • acceptance_rate
      (chain, draw)
      float64
      0.7452 0.4529 ... 0.8597 0.9824

      array([[0.74516657, 0.45290931, 0.9685891 , ..., 1.        , 0.81275417,
              0.74543394],
             [0.35366975, 0.84103863, 0.9864095 , ..., 0.77299459, 0.97959445,
              1.        ],
             [0.88737108, 0.58326352, 0.98718512, ..., 0.97927302, 0.21914696,
              0.96483155],
             [0.8747444 , 0.81408187, 0.8800982 , ..., 1.        , 0.85965816,
              0.98239012]])

    • diverging
      (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]])

    • energy
      (chain, draw)
      float64
      33.97 36.99 34.41 ... 33.44 33.49

      array([[33.97259792, 36.9913773 , 34.4085046 , ..., 33.35895831,
              33.5887212 , 33.7520542 ],
             [35.10040915, 33.31122128, 33.14971609, ..., 33.42777433,
              33.81221173, 33.2367959 ],
             [33.11697833, 33.87341633, 32.88778339, ..., 33.21889799,
              34.12835248, 32.97094125],
             [33.38605933, 33.78022511, 34.37503189, ..., 33.33355514,
              33.43569456, 33.49374829]])

    • energy_error
      (chain, draw)
      float64
      0.2941 -0.4251 ... 0.1512 -0.2058

      array([[ 0.2941475 , -0.42505719,  0.09897285, ..., -0.21767696,
              -0.01082192,  0.0234105 ],
             [ 0.        ,  0.17311769, -0.19261355, ...,  0.25748323,
               0.0584438 , -0.31266731],
             [ 0.01875866, -0.07102908,  0.0128977 , ..., -0.34242052,
               0.        ,  0.05604317],
             [ 0.11361917,  0.20569434,  0.12772179, ..., -0.17976445,
               0.15122046, -0.20577766]])

    • index_in_trajectory
      (chain, draw)
      int64
      -1 2 2 3 -3 -1 -1 ... -1 0 1 -1 1 2

      array([[-1,  2,  2, ..., -1, -2, -2],
             [ 0, -1, -3, ...,  1,  1, -1],
             [-2,  2, -1, ...,  1,  0, -1],
             [-2, -1,  1, ..., -1,  1,  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]])

    • lp
      (chain, draw)
      float64
      -33.9 -34.05 ... -33.42 -32.95

      array([[-33.89695623, -34.05219472, -34.31518136, ..., -33.01049785,
              -33.08536449, -33.01197245],
             [-32.89479742, -33.30839416, -32.86404386, ..., -33.42730218,
              -33.54794717, -32.87424822],
             [-32.93082919, -32.86289244, -32.88778282, ..., -32.86338325,
              -32.86338325, -32.97092543],
             [-33.14053858, -33.73395094, -34.12957899, ..., -32.99636341,
              -33.41757933, -32.94911324]])

    • max_energy_error
      (chain, draw)
      float64
      0.2941 2.003 ... 0.1512 -0.2058

      array([[ 0.2941475 ,  2.00284402, -0.53497001, ..., -0.21767696,
               0.37660698,  0.52560849],
             [ 1.89329128,  0.17311769, -0.19261355, ...,  0.25748323,
               0.0584438 , -0.31402413],
             [ 0.18464581,  1.13803139,  0.0128977 , ..., -0.34242052,
               1.51801273,  0.05604317],
             [ 0.20780599,  0.20569434,  0.12772179, ..., -0.17976445,
               0.15122046, -0.20577766]])

    • n_steps
      (chain, draw)
      float64
      1.0 3.0 3.0 3.0 ... 1.0 1.0 1.0 3.0

      array([[1., 3., 3., ..., 1., 3., 3.],
             [3., 1., 3., ..., 1., 3., 3.],
             [3., 3., 1., ..., 3., 1., 3.],
             [3., 1., 1., ..., 1., 1., 3.]])

    • perf_counter_diff
      (chain, draw)
      float64
      0.0001378 7.529e-05 ... 0.0001853

      array([[1.37834111e-04, 7.52909109e-05, 3.22791981e-04, ...,
              7.34159257e-05, 2.44708033e-04, 1.81333162e-04],
             [7.60001130e-05, 3.16668302e-05, 7.17910007e-05, ...,
              7.68750906e-05, 1.60875032e-04, 1.49334082e-04],
             [1.78915914e-04, 1.51125016e-04, 6.97919168e-05, ...,
              1.43415993e-04, 7.02079851e-05, 1.41290948e-04],
             [1.80375064e-04, 9.42500774e-05, 8.28751363e-05, ...,
              7.30000902e-05, 7.29158055e-05, 1.85291981e-04]])

    • perf_counter_start
      (chain, draw)
      float64
      1.548e+06 1.548e+06 ... 1.548e+06

      array([[1547567.37198667, 1547567.37218238, 1547567.37246692, ...,
              1547567.7886445 , 1547567.78904771, 1547567.78935521],
             [1547567.40217446, 1547567.40227929, 1547567.40242625, ...,
              1547567.85699087, 1547567.85762037, 1547567.85783754],
             [1547567.41182413, 1547567.41206429, 1547567.41227017, ...,
              1547567.973906  , 1547567.97410187, 1547567.97422325],
             [1547567.5908475 , 1547567.59191479, 1547567.59278812, ...,
              1547568.08235779, 1547568.08254775, 1547568.08275554]])

    • process_time_diff
      (chain, draw)
      float64
      0.000111 7.5e-05 ... 0.000184

      array([[1.11e-04, 7.50e-05, 1.95e-04, ..., 7.40e-05, 1.88e-04, 1.64e-04],
             [7.50e-05, 3.20e-05, 7.10e-05, ..., 7.60e-05, 1.61e-04, 1.50e-04],
             [1.78e-04, 1.52e-04, 6.90e-05, ..., 1.43e-04, 7.00e-05, 1.42e-04],
             [1.80e-04, 9.40e-05, 8.40e-05, ..., 7.20e-05, 7.30e-05, 1.84e-04]])

    • 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]])

    • 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]])

    • step_size
      (chain, draw)
      float64
      1.702 1.702 1.702 ... 1.317 1.317

      array([[1.70234384, 1.70234384, 1.70234384, ..., 1.70234384, 1.70234384,
              1.70234384],
             [1.17465906, 1.17465906, 1.17465906, ..., 1.17465906, 1.17465906,
              1.17465906],
             [1.08848323, 1.08848323, 1.08848323, ..., 1.08848323, 1.08848323,
              1.08848323],
             [1.31729389, 1.31729389, 1.31729389, ..., 1.31729389, 1.31729389,
              1.31729389]])

    • step_size_bar
      (chain, draw)
      float64
      1.415 1.415 1.415 ... 1.327 1.327

      array([[1.41481789, 1.41481789, 1.41481789, ..., 1.41481789, 1.41481789,
              1.41481789],
             [1.33229145, 1.33229145, 1.33229145, ..., 1.33229145, 1.33229145,
              1.33229145],
             [1.50093681, 1.50093681, 1.50093681, ..., 1.50093681, 1.50093681,
              1.50093681],
             [1.32705532, 1.32705532, 1.32705532, ..., 1.32705532, 1.32705532,
              1.32705532]])

    • tree_depth
      (chain, draw)
      int64
      1 2 2 2 2 2 2 1 ... 2 2 1 1 1 1 1 2

      array([[1, 2, 2, ..., 1, 2, 2],
             [2, 1, 2, ..., 1, 2, 2],
             [2, 2, 1, ..., 2, 1, 2],
             [2, 1, 1, ..., 1, 1, 2]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             990, 991, 992, 993, 994, 995, 996, 997, 998, 999],
            dtype='int64', name='draw', length=1000))

  • Attributes: (8)

    created_at :

    2025-03-12T12:51:55.673444+00:00

    arviz_version :

    0.19.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    sampling_time :

    1.81168794631958

    tuning_steps :

    1000

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  
• observed_data

  <xarray.Dataset> Size: 800B
  Dimensions:  (__obs__: 50)
  Coordinates:
    * __obs__  (__obs__) int64 400B 0 1 2 3 4 5 6 7 8 ... 42 43 44 45 46 47 48 49
  Data variables:
      ctoss    (__obs__) int64 400B 1 1 0 0 1 0 1 1 1 1 1 ... 0 1 0 1 0 1 0 1 1 0
  Attributes:
      created_at:                  2025-03-12T12:51:55.675308+00:00
      arviz_version:               0.19.0
      inference_library:           pymc
      inference_library_version:   5.21.1
      modeling_interface:          bambi
      modeling_interface_version:  0.15.0

  xarray.Dataset

  • Dimensions:
    • __obs__: 50
  • Coordinates: (1)
    • __obs__
      (__obs__)
      int64
      0 1 2 3 4 5 6 ... 44 45 46 47 48 49

      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])

  • Data variables: (1)
    • ctoss
      (__obs__)
      int64
      1 1 0 0 1 0 1 1 ... 0 1 0 1 0 1 1 0

      array([1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1,
             1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1,
             0, 1, 0, 1, 1, 0])

  • Indexes: (1)
    • __obs__
      PandasIndex

      PandasIndex(Index([ 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],
            dtype='int64', name='__obs__'))

  • Attributes: (6)

    created_at :

    2025-03-12T12:51:55.675308+00:00

    arviz_version :

    0.19.0

    inference_library :

    pymc

    inference_library_version :

    5.21.1

    modeling_interface :

    bambi

    modeling_interface_version :

    0.15.0

  
  

In [66]:

type(idata1)

type(idata1)

Out[66]:

arviz.data.inference_data.InferenceData

In [67]:

idata1.groups()

idata1.groups()

Out[67]:

['posterior', 'sample_stats', 'observed_data']

Groups are Dataset objects¶

In [68]:

post1 = idata1["posterior"]
type(post1)

post1 = idata1["posterior"] type(post1)

Out[68]:

xarray.core.dataset.Dataset

In [69]:

# post1

# post1

In [70]:

post1.coords

post1.coords

Out[70]:

Coordinates:
  * chain    (chain) int64 32B 0 1 2 3
  * draw     (draw) int64 8kB 0 1 2 3 4 5 6 7 ... 993 994 995 996 997 998 999

In [71]:

post1.data_vars

post1.data_vars

Out[71]:

Data variables:
    P        (chain, draw) float64 32kB 0.7612 0.5686 0.7762 ... 0.7388 0.6457

Variables are DataArray objects¶

In [72]:

Ps = post1["P"]  # == idata1["posterior"]["P"]
type(Ps)

Ps = post1["P"] # == idata1["posterior"]["P"] type(Ps)

Out[72]:

xarray.core.dataarray.DataArray

In [73]:

# Ps

# Ps

In [74]:

Ps.shape

Ps.shape

Out[74]:

(4, 1000)

In [75]:

Ps.name

Ps.name

Out[75]:

'P'

In [76]:

Ps.to_dataframe()

Ps.to_dataframe()

Out[76]:

		P
chain	draw
0	0	0.761228
	1	0.568612
	2	0.776170
	3	0.675721
	4	0.749423
...	...	...
3	995	0.767060
	996	0.743217
	997	0.706058
	998	0.738807
	999	0.645698

4000 rows × 1 columns

In [77]:

Ps.values

Ps.values

Out[77]:

array([[0.76122756, 0.56861221, 0.77616961, ..., 0.70772348, 0.62878475,
        0.70789234],
       [0.68937002, 0.73226925, 0.67623114, ..., 0.60182373, 0.59438488,
        0.6632119 ],
       [0.69674422, 0.67356423, 0.65844755, ..., 0.67517019, 0.67517019,
        0.64238598],
       [0.72020439, 0.75444132, 0.7699058 , ..., 0.70605793, 0.73880719,
        0.64569808]])

In [78]:

Ps.values.flatten()

Ps.values.flatten()

Out[78]:

array([0.76122756, 0.56861221, 0.77616961, ..., 0.70605793, 0.73880719,
       0.64569808])

MCMC diagnostics¶

Trace plots¶

There are several Arviz plots we can use to check if the Markov Chain Monte Carlo chains were sampling from the posterior as expected, or ...

In [79]:

az.plot_trace(idata2);

az.plot_trace(idata2);

Rank plots¶

In [80]:

az.plot_rank(idata2);

az.plot_rank(idata2);

? Autocorrelation plot ?¶

https://python.arviz.org/en/stable/api/generated/arviz.plot_autocorr.html

In [81]:

az.plot_autocorr(idata2, combined=True, max_lag=20);

az.plot_autocorr(idata2, combined=True, max_lag=20);

Diagnostic summary¶

In [82]:

az.summary(idata2, kind="stats")

az.summary(idata2, kind="stats")

Out[82]:

	mean	sd	hdi_3%	hdi_97%
M	107.727	2.715	102.592	112.844

In [83]:

az.summary(idata2, kind="diagnostics")

az.summary(idata2, kind="diagnostics")

Out[83]:

	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
M	0.047	0.034	3290.0	5181.0	1.0

Bayesian workflow¶

See also:

• chp_09.ipynb
• Section7-Bayesian_Workflow.ipynb

Prior predictive checks¶

In [84]:

idata2_pri = mod2.prior_predictive(draws=50, random_seed=45)
az.plot_ppc(idata2_pri, group="prior");

idata2_pri = mod2.prior_predictive(draws=50, random_seed=45) az.plot_ppc(idata2_pri, group="prior");

Sampling: [M, iq]

In [85]:

idata2_pri['prior']["mu"].shape

idata2_pri['prior']["mu"].shape

Out[85]:

(1, 50, 30)

In [86]:

idata2_pri['prior_predictive']["iq"].shape

idata2_pri['prior_predictive']["iq"].shape

Out[86]:

(1, 50, 30)

Posterior predictive checks¶

In [87]:

# Randomly sample 200 from all available posterior mean
# draws_subset = np.random.choice(idata2["posterior"]["draw"].values, 50, replace=False)
draws_subset = np.random.choice(2000, 50, replace=False)
idata2_rep = idata2.sel(draw=draws_subset)
mod2.predict(idata2_rep, kind="response")
az.plot_ppc(idata2_rep, group="posterior")

# Randomly sample 200 from all available posterior mean # draws_subset = np.random.choice(idata2["posterior"]["draw"].values, 50, replace=False) draws_subset = np.random.choice(2000, 50, replace=False) idata2_rep = idata2.sel(draw=draws_subset) mod2.predict(idata2_rep, kind="response") az.plot_ppc(idata2_rep, group="posterior")

Out[87]:

<Axes: xlabel='iq'>

Fitting model to synthetic data¶

In [88]:

np.random.seed(46)
from scipy.stats import norm
fakeiqs = norm(loc=110, scale=15).rvs(30)
fakeiqs = pd.DataFrame({"iq":fakeiqs})
fakeiqs["iq"].mean()

np.random.seed(46) from scipy.stats import norm fakeiqs = norm(loc=110, scale=15).rvs(30) fakeiqs = pd.DataFrame({"iq":fakeiqs}) fakeiqs["iq"].mean()

Out[88]:

109.7132680142492

In [89]:

#######################################################
mod2f = bmb.Model(formula="iq ~ 1", family="gaussian",
                  priors=priors2, data=fakeiqs)
mod2f.set_alias({"Intercept": "M"})
idata2f = mod2f.fit(random_seed=42)

####################################################### mod2f = bmb.Model(formula="iq ~ 1", family="gaussian", priors=priors2, data=fakeiqs) mod2f.set_alias({"Intercept": "M"}) idata2f = mod2f.fit(random_seed=42)

Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [M]

Output()

Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 2 seconds.

In [90]:

az.summary(idata2f, kind="stats")

az.summary(idata2f, kind="stats")

Out[90]:

	mean	sd	hdi_3%	hdi_97%
M	109.65	2.736	104.645	114.719

In [91]:

az.plot_posterior(idata2f);

az.plot_posterior(idata2f);

Sensitivity analysis¶

In [92]:

# TODO: reproduce sensitivity analysis from Secion 5.1

# TODO: reproduce sensitivity analysis from Secion 5.1

Iterative model building¶

Discussion¶

Other types of priors¶

Conjugate priors¶

Uninformative priors¶

Software for Bayesian inference¶

Computational cost¶

Reporting Bayesian results¶

Model predictions (BONUS TOPIC)¶

Coin toss predictions¶

Let's generate some observations form the posterior predictive distribution.

In [93]:

idata1_pred = idata1.copy()
# MAYBE: alt notation _rep instead of _pred ?
mod1.predict(idata1_pred, kind="response")
ctosses_pred = az.extract(idata1_pred,
                          group="posterior_predictive",
                          var_names=["ctoss"]).values
numheads_pred = ctosses_pred.sum(axis=0)

sns.histplot(x=numheads_pred, bins=range(15,50));

idata1_pred = idata1.copy() # MAYBE: alt notation _rep instead of _pred ? mod1.predict(idata1_pred, kind="response") ctosses_pred = az.extract(idata1_pred, group="posterior_predictive", var_names=["ctoss"]).values numheads_pred = ctosses_pred.sum(axis=0) sns.histplot(x=numheads_pred, bins=range(15,50));

IQ predictions¶

In [94]:

## Can't use Bambi predict because of issue
## https://github.com/bambinos/bambi/issues/850
# idata2_pred = mod2.predict(idata2, kind="response", inplace=False)
# preds2 = az.extract(idata2_pred, group="posterior", var_names=["mu"]).values.flatten()

## Can't use Bambi predict because of issue ## https://github.com/bambinos/bambi/issues/850 # idata2_pred = mod2.predict(idata2, kind="response", inplace=False) # preds2 = az.extract(idata2_pred, group="posterior", var_names=["mu"]).values.flatten()

In [95]:

from scipy.stats import norm

sigma = 15  # known population standard deviation

iq_preds = []
np.random.seed(42)
for i in range(1000):
    mu_post = np.random.choice(postM)
    iq_pred = norm(loc=mu_post, scale=sigma).rvs(1)[0]
    iq_preds.append(iq_pred)

sns.histplot(x=iq_preds);

from scipy.stats import norm sigma = 15 # known population standard deviation iq_preds = [] np.random.seed(42) for i in range(1000): mu_post = np.random.choice(postM) iq_pred = norm(loc=mu_post, scale=sigma).rvs(1)[0] iq_preds.append(iq_pred) sns.histplot(x=iq_preds);

Exercises¶

TODO: repeat exercises form Sec 5.1 using Bambi and ArviZ

Exercise A¶

Reproduce the analysis of the IQ scores data from Example 2, but this time use a model with Bambi default priors.

a) Plot the priors using plot_priors.
b) Compute the posterior mean, std, and a 90% hdi.
c) Compare your result in (b) to the results we obtained from Example 2.

In [96]:

mod2d = bmb.Model("iq ~ 1", family="gaussian", data=iqs)
mod2d.set_alias({"Intercept": "M"})
mod2d.build()
mod2d.plot_priors();

mod2d = bmb.Model("iq ~ 1", family="gaussian", data=iqs) mod2d.set_alias({"Intercept": "M"}) mod2d.build() mod2d.plot_priors();

Sampling: [M, sigma]

In [97]:

idata2d = mod2d.fit(draws=2000, random_seed=42)
az.summary(idata2d, var_names="M", kind="stats", hdi_prob=0.9)

idata2d = mod2d.fit(draws=2000, random_seed=42) az.summary(idata2d, var_names="M", kind="stats", hdi_prob=0.9)

Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [sigma, M]

Output()

Sampling 4 chains for 1_000 tune and 2_000 draw iterations (4_000 + 8_000 draws total) took 3 seconds.

Out[97]:

	mean	sd	hdi_5%	hdi_95%
M	107.789	3.081	102.654	112.704

In [98]:

az.summary(idata2, kind="stats", hdi_prob=0.9)

az.summary(idata2, kind="stats", hdi_prob=0.9)

Out[98]:

	mean	sd	hdi_5%	hdi_95%
M	107.727	2.715	103.253	112.128

In [99]:

az.plot_forest(data=[idata2, idata2d],
               model_names=["Example 2 prior", "Default priors"],
               var_names="M", combined=True, figsize=(7,2));

az.plot_forest(data=[idata2, idata2d], model_names=["Example 2 prior", "Default priors"], var_names="M", combined=True, figsize=(7,2));

Exercise B¶

Repeat the analysis in Example 2, but this time use the exponential prior $\text{Expon}(\lambda=0.1)$ on the parameter sigma. Calculate:

a) Plot the priors using plot_priors.
b) Compute posterior mean of M and sigma.
c) Compute the posterior median of M and sigma.
d) Compute 90% credible interval for M and sigma.

In [100]:

import bambi as bmb
priors2e = {
    "Intercept": bmb.Prior("Normal", mu=100, sigma=40),
    "sigma": bmb.Prior("Exponential", lam=0.1),
}

mod2e = bmb.Model(formula="iq ~ 1",
                  family="gaussian",
                  link="identity",
                  priors=priors2e,
                  data=iqs)
mod2e.set_alias({"Intercept": "M"})

import bambi as bmb priors2e = { "Intercept": bmb.Prior("Normal", mu=100, sigma=40), "sigma": bmb.Prior("Exponential", lam=0.1), } mod2e = bmb.Model(formula="iq ~ 1", family="gaussian", link="identity", priors=priors2e, data=iqs) mod2e.set_alias({"Intercept": "M"})

In [101]:

mod2e.build()
mod2e.plot_priors();

mod2e.build() mod2e.plot_priors();

Sampling: [M, sigma]

In [102]:

# idata2e = mod2e.fit(random_seed=42)
# az.plot_posterior(idata2e)

# idata2e = mod2e.fit(random_seed=42) # az.plot_posterior(idata2e)

In [103]:

# JOINT PLOT
# M_samples = idata2e.posterior['M'].values.flatten()
# sigma_samples = idata2e.posterior['sigma'].values.flatten()
# sns.kdeplot(x=M_samples, y=sigma_samples, fill=True)
# plt.xlabel('M')
# plt.ylabel('Sigma')
# plt.title('Density Plot of Mean vs Sigma')

# JOINT PLOT # M_samples = idata2e.posterior['M'].values.flatten() # sigma_samples = idata2e.posterior['sigma'].values.flatten() # sns.kdeplot(x=M_samples, y=sigma_samples, fill=True) # plt.xlabel('M') # plt.ylabel('Sigma') # plt.title('Density Plot of Mean vs Sigma')

Exercise C¶

In [ ]:

Links¶

TODO

In [ ]:

In [ ]:

In [ ]:

In [ ]: