Implementing a RandomVariable Distribution#

This guide provides an overview on how to implement a distribution for PyMC version >=4.0.0. It is designed for developers who wish to add a new distribution to the library. Users will not be aware of all this complexity and should instead make use of helper methods such as ~pymc.DensityDist.

PyMC Distribution builds on top of Aesara’s RandomVariable, and implements logp, logcdf and moment methods as well as other initialization and validation helpers. Most notably shape/dims/observed kwargs, alternative parametrizations, and default transform.

Here is a summary check-list of the steps needed to implement a new distribution. Each section will be expanded below:

  1. Creating a new RandomVariable Op

  2. Implementing the corresponding Distribution class

  3. Adding tests for the new RandomVariable

  4. Adding tests for logp / logcdf and moment methods

  5. Documenting the new Distribution.

This guide does not attempt to explain the rationale behind the Distributions current implementation, and details are provided only insofar as they help to implement new “standard” distributions.

1. Creating a new RandomVariable Op#

RandomVariable are responsible for implementing the random sampling methods, which in version 3 of PyMC used to be one of the standard Distribution methods, alongside logp and logcdf. The RandomVariable is also responsible for parameter broadcasting and shape inference.

Before creating a new RandomVariable make sure that it is not already offered in the NumPy library. If it is, it should be added to the Aesara library first and then imported into the PyMC library.

In addition, it might not always be necessary to implement a new RandomVariable. For example if the new Distribution is just a special parametrization of an existing Distribution. This is the case of the OrderedLogistic and OrderedProbit, which are just special parametrizations of the Categorical distribution.

The following snippet illustrates how to create a new RandomVariable:

from aesara.tensor.var import TensorVariable
from aesara.tensor.random.op import RandomVariable
from typing import List, Tuple

# Create your own `RandomVariable`...
class BlahRV(RandomVariable):
    name: str = "blah"

    # Provide the minimum number of (output) dimensions for this RV
    # (e.g. `0` for a scalar, `1` for a vector, etc.)
    ndim_supp: int = 0

    # Provide the number of (input) dimensions for each parameter of the RV
    # (e.g. if there's only one vector parameter, `[1]`; for two parameters,
    # one a matrix and the other a scalar, `[2, 0]`; etc.)
    ndims_params: List[int] = [0, 0]

    # The NumPy/Aesara dtype for this RV (e.g. `"int32"`, `"int64"`).
    # The standard in the library is `"int64"` for discrete variables
    # and `"floatX"` for continuous variables
    dtype: str = "floatX"

    # A pretty text and LaTeX representation for the RV
    _print_name: Tuple[str, str] = ("blah", "\\operatorname{blah}")

    # If you want to add a custom signature and default values for the
    # parameters, do it like this. Otherwise this can be left out.
    def __call__(self, loc=0.0, scale=1.0, **kwargs) -> TensorVariable:
        return super().__call__(loc, scale, **kwargs)

    # This is the Python code that produces samples.  Its signature will always
    # start with a NumPy `RandomState` object, then the distribution
    # parameters, and, finally, the size.
    # This is effectively the PyMC >=4.0 replacement for `Distribution.random`.
    def rng_fn(
        rng: np.random.RandomState,
        loc: np.ndarray,
        scale: np.ndarray,
        size: Tuple[int, ...],
    ) -> np.ndarray:
        return scipy.stats.blah.rvs(loc, scale, random_state=rng, size=size)

# Create the actual `RandomVariable` `Op`...
blah = BlahRV()

Some important things to keep in mind:

  1. Everything inside the rng_fn method is pure Python code (as are the inputs) and should not make use of other Aesara symbolic ops. The random method should make use of the rng which is a NumPy RandomState, so that samples are reproducible.

  2. Non-default RandomVariable dimensions will end up in the rng_fn via the size kwarg. The rng_fn will have to take this into consideration for correct output. size is the specification used by NumPy and SciPy and works like PyMC shape for univariate distributions, but is different for multivariate distributions. For multivariate distributions the size excludes the ndim_supp support dimensions, whereas the shape of the resulting TensorVariabe or ndarray includes the support dimensions. For more context check The dimensionality notebook.

  3. Aesara tries to infer the output shape of the RandomVariable (given a user-specified size) by introspection of the ndim_supp and ndim_params attributes. However, the default method may not work for more complex distributions. In that case, custom _supp_shape_from_params (and less probably, _infer_shape) should also be implemented in the new RandomVariable class. One simple example is seen in the DirichletMultinomialRV where it was necessary to specify the rep_param_idx so that the default_supp_shape_from_params helper method can do its job. In more complex cases, it may not suffice to use this default helper. This could happen for instance if the argument values determined the support shape of the distribution, as happens in the ~pymc.distributions.multivarite._LKJCholeskyCovRV.

  4. It’s okay to use the rng_fn classmethods of other Aesara and PyMC RandomVariables inside the new rng_fn. For example if you are implementing a negative HalfNormal RandomVariable, your rng_fn can simply return - halfnormal.rng_fn(rng, scale, size).

Note: In addition to size, the PyMC API also provides shape, dims and observed as alternatives to define a distribution dimensionality, but this is taken care of by Distribution, and should not require any extra changes.

For a quick test that your new RandomVariable Op is working, you can call the Op with the necessary parameters and then call eval() on the returned object:

# blah = aesara.tensor.random.uniform in this example
blah([0, 0], [1, 2], size=(10, 2)).eval()

# array([[0.83674527, 0.76593773],
#    [0.00958496, 1.85742402],
#    [0.74001876, 0.6515534 ],
#    [0.95134629, 1.23564938],
#    [0.41460156, 0.33241175],
#    [0.66707807, 1.62134924],
#    [0.20748312, 0.45307477],
#    [0.65506507, 0.47713784],
#    [0.61284429, 0.49720329],
#    [0.69325978, 0.96272673]])

2. Inheriting from a PyMC base Distribution class#

After implementing the new RandomVariable Op, it’s time to make use of it in a new PyMC Distribution. PyMC >=4.0.0 works in a very functional way, and the distribution classes are there mostly to facilitate porting the PyMC3 v3.x code to PyMC >=4.0.0, add PyMC API features and keep related methods organized together. In practice, they take care of:

  1. Linking (Dispatching) an rv_op class with the corresponding moment, logp and logcdf methods.

  2. Defining a standard transformation (for continuous distributions) that converts a bounded variable domain (e.g., positive line) to an unbounded domain (i.e., the real line), which many samplers prefer.

  3. Validating the parametrization of a distribution and converting non-symbolic inputs (i.e., numeric literals or NumPy arrays) to symbolic variables.

  4. Converting multiple alternative parametrizations to the standard parametrization that the RandomVariable is defined in terms of.

Here is how the example continues:

import aesara.tensor as at
from pymc.aesaraf import floatX, intX
from pymc.distributions.continuous import PositiveContinuous
from pymc.distributions.dist_math import check_parameters
from pymc.distributions.shape_utils import rv_size_is_none

# Subclassing `PositiveContinuous` will dispatch a default `log` transformation
class Blah(PositiveContinuous):
    # This will be used by the metaclass `DistributionMeta` to dispatch the
    # class `logp` and `logcdf` methods to the `blah` `Op` defined in the last line of the code above.
    rv_op = blah

    # dist() is responsible for returning an instance of the rv_op.
    # We pass the standard parametrizations to super().dist
    def dist(cls, param1, param2=None, alt_param2=None, **kwargs):
        param1 = at.as_tensor_variable(intX(param1))
        if param2 is not None and alt_param2 is not None:
            raise ValueError("Only one of param2 and alt_param2 is allowed.")
        if alt_param2 is not None:
            param2 = 1 / alt_param2
        param2 = at.as_tensor_variable(floatX(param2))

        # The first value-only argument should be a list of the parameters that
        # the rv_op needs in order to be instantiated
        return super().dist([param1, param2], **kwargs)

    # moment returns a symbolic expression for the stable moment from which to start sampling
    # the variable, given the implicit `rv`, `size` and `param1` ... `paramN`.
    # This is typically a "representative" point such as the the mean or mode.
    def moment(rv, size, param1, param2):
        moment, _ = at.broadcast_arrays(param1, param2)
        if not rv_size_is_none(size):
            moment = at.full(size, moment)
        return moment

    # Logp returns a symbolic expression for the elementwise log-pdf or log-pmf evaluation
    # of the variable given the `value` of the variable and the parameters `param1` ... `paramN`.
    def logp(value, param1, param2):
        logp_expression = value * (param1 + at.log(param2))

        # A switch is often used to enforce the distribution support domain
        bounded_logp_expression = at.switch(
   >= 0),

        # We use `check_parameters` for parameter validation. After the default expression,
        # multiple comma-separated symbolic conditions can be added.
        # Whenever a bound is invalidated, the returned expression raises an error
        # with the message defined in the optional `msg` keyword argument.
        return check_parameters(
            param2 >= 0,
            msg="param2 >= 0",

    # logcdf works the same way as logp. For bounded variables, it is expected to return
    # `-inf` for values below the domain start and `0` for values above the domain end.
    def logcdf(value, param1, param2):

Some notes:

  1. A distribution should at the very least inherit from Discrete or Continuous. For the latter, more specific subclasses exist: PositiveContinuous, UnitContinuous, BoundedContinuous, CircularContinuous, SimplexContinuous, which specify default transformations for the variables. If you need to specify a one-time custom transform you can also create a _default_transform dispatch function as is done for the LKJCholeskyCov.

  2. If a distribution does not have a corresponding rng_fn implementation, a RandomVariable should still be created to raise a NotImplementedError. This is, for example, the case in Flat. In this case it will be necessary to provide a moment method, because without a rng_fn, PyMC can’t fall back to a random draw to use as an initial point for MCMC.

  3. As mentioned above, PyMC >=4.0.0 works in a very functional way, and all the information that is needed in the logp and logcdf methods is expected to be “carried” via the RandomVariable inputs. You may pass numerical arguments that are not strictly needed for the rng_fn method but are used in the logp and logcdf methods. Just keep in mind whether this affects the correct shape inference behavior of the RandomVariable. If specialized non-numeric information is needed you might need to define your custom_logp and _logcdf Dispatching functions, but this should be done as a last resort.

  4. The logcdf method is not a requirement, but it’s a nice plus!

  5. Currently, only one moment is supported in the moment method, and probably the “higher-order” one is the most useful (that is mean > median > mode)… You might need to truncate the moment if you are dealing with a discrete distribution.

  6. When creating the moment method, be careful with size != None and broadcast properly also based on parameters that are not necessarily used to calculate the moment. For example, the sigma in pm.Normal.dist(mu=0, sigma=np.arange(1, 6)) is irrelevant for the moment, but may nevertheless inform about the shape. In this case, the moment should return [mu, mu, mu, mu, mu].

For a quick check that things are working you can try the following:

import pymc as pm
from pymc.distributions.distribution import moment

# pm.blah = pm.Normal in this example
blah = pm.blah.dist(mu=0, sigma=1)

# Test that the returned blah_op is still working fine
# array(-1.01397228)

# Test the moment method
# array(0.)

# Test the logp method
pm.logp(blah, [-0.5, 1.5]).eval()
# array([-1.04393853, -2.04393853])

# Test the logcdf method
pm.logcdf(blah, [-0.5, 1.5]).eval()
# array([-1.17591177, -0.06914345])

3. Adding tests for the new RandomVariable#

Tests for new RandomVariables are mostly located in pymc/tests/distributions/test_*.py. Most tests can be accommodated by the default BaseTestDistributionRandom class, which provides default tests for checking:

  1. Expected inputs are passed to the rv_op by the dist classmethod, via check_pymc_params_match_rv_op

  2. Expected (exact) draws are being returned, via check_pymc_draws_match_reference

  3. Shape variable inference is correct, via check_rv_size

from pymc.tests.distributions.util import BaseTestDistributionRandom, seeded_scipy_distribution_builder

class TestBlah(BaseTestDistributionRandom):

    pymc_dist = pm.Blah
    # Parameters with which to test the blah pymc Distribution
    pymc_dist_params = {"param1": 0.25, "param2": 2.0}
    # Parameters that are expected to have passed as inputs to the RandomVariable op
    expected_rv_op_params = {"param1": 0.25, "param2": 2.0}
    # If the new `RandomVariable` is simply calling a `numpy`/`scipy` method,
    # we can make use of `seeded_[scipy|numpy]_distribution_builder` which
    # will prepare a seeded reference distribution for us.
    reference_dist_params = {"mu": 0.25, "loc": 2.0}
    reference_dist = seeded_scipy_distribution_builder("blah")
    tests_to_run = [

Additional tests should be added for each optional parametrization of the distribution. In this case it’s enough to include the test check_pymc_params_match_rv_op since only this differs.

Make sure the tested alternative parameter value would lead to a different value for the associated default parameter. For instance, if it’s just the inverse, testing with 1.0 is not very informative, since the conversion would return 1.0 as well, and we can’t be (as) sure that is working correctly.

class TestBlahAltParam2(BaseTestDistributionRandom):

    pymc_dist = pm.Blah
    # param2 is equivalent to 1 / alt_param2
    pymc_dist_params = {"param1": 0.25, "alt_param2": 4.0}
    expected_rv_op_params = {"param1": 0.25, "param2": 2.0}
    tests_to_run = ["check_pymc_params_match_rv_op"]

Custom tests can also be added to the class as is done for the TestFlat.

Note on check_rv_size test:#

Custom input sizes (and expected output shapes) can be defined for the check_rv_size test, by adding the optional class attributes sizes_to_check and sizes_expected:

sizes_to_check = [None, (1), (2, 3)]
sizes_expected = [(3,), (1, 3), (2, 3, 3)]
tests_to_run = ["check_rv_size"]

This is usually needed for Multivariate distributions. You can see an example in TestDirichlet.

Notes on check_pymcs_draws_match_reference test#

The check_pymcs_draws_match_reference is a very simple test for the equality of draws from the RandomVariable and the exact same python function, given the same inputs and random seed. A small number (size=15) is checked. This is not supposed to be a test for the correctness of the random number generator. The latter kind of test (if warranted) can be performed with the aid of pymc_random and pymc_random_discrete methods, which will perform an expensive statistical comparison between the RandomVariable.rng_fn and a reference Python function. This kind of test only makes sense if there is a good independent generator reference (i.e., not just the same composition of NumPy / SciPy calls that is done inside rng_fn).

Finally, when your rng_fn is doing something more than just calling a NumPy or SciPy method, you will need to set up an equivalent seeded function with which to compare for the exact draws (instead of relying on seeded_[scipy|numpy]_distribution_builder). You can find an example in TestWeibull, whose rng_fn returns beta * np.random.weibull(alpha, size=size).

4. Adding tests for the logp / logcdf methods#

Tests for the logp and logcdf mostly make use of the helpers check_logp, check_logcdf, and check_selfconsistency_discrete_logcdf implemented in ~pymc.tests.distributions.util

from pymc.tests.distributions.util import check_logp, check_logcdf, Domain
from pymc.tests.helpers import select_by_precision

R = Domain([-np.inf, -2.1, -1, -0.01, 0.0, 0.01, 1, 2.1, np.inf])
Rplus = Domain([0, 0.01, 0.1, 0.9, 0.99, 1, 1.5, 2, 100, np.inf])

def test_blah():

      # Domain of the distribution values
      # Domains of the distribution parameters
      paramdomains={"mu": R, "sigma": Rplus},
      # Reference scipy (or other) logp function
      scipy_logp = lambda value, mu, sigma: sp.norm.logpdf(value, mu, sigma),
      # Number of decimal points expected to match between the pymc and reference functions
      decimal=select_by_precision(float64=6, float32=3),
      # Maximum number of combinations of domain * paramdomains to test

      paramdomains={"mu": R, "sigma": Rplus},
      scipy_logcdf=lambda value, mu, sigma: sp.norm.logcdf(value, mu, sigma),
      decimal=select_by_precision(float64=6, float32=1),

These methods will perform a grid evaluation on the combinations of domain and paramdomains values, and check that the PyMC methods and the reference functions match. There are a couple of details worth keeping in mind:

  1. By default, the first and last values (edges) of the Domain are not compared (they are used for other things). If it is important to test the edge of the Domain, the edge values can be repeated. This is done by the Bool: Bool = Domain([0, 0, 1, 1], "int64")

  2. There are some default domains (such as R and Rplus) that you can use for testing your new distribution, but it’s also perfectly fine to create your own domains inside the test function if there is a good reason for it (e.g., when the default values lead too many extreme unlikely combinations that are not very informative about the correctness of the implementation).

  3. By default, a random subset of 100 param x paramdomain combinations is tested, to keep the test runtime under control. When testing your shiny new distribution, you can temporarily set n_samples=-1 to force all combinations to be tested. This is important to avoid your PR leading to surprising failures in future runs whenever some bad combinations of parameters are randomly tested.

  4. On GitHub some tests run twice, under the aesara.config.floatX flags of "float64" and "float32". However, the reference Python functions will run in a pure “float64” environment, which means the reference and the PyMC results can diverge quite a lot (e.g., underflowing to -np.inf for extreme parameters). You should therefore make sure you test locally in both regimes. A quick and dirty way of doing this is to temporarily add aesara.config.floatX = "float32" at the very top of file, immediately after import aesara. Remember to set n_samples=-1 as well to test all combinations. The test output will show what exact parameter values lead to a failure. If you are confident that your implementation is correct, you may opt to tweak the decimal precision with select_by_precision, or adjust the tested Domain values. In extreme cases, you can mark the test with a conditional xfail (if only one of the sub-methods is failing, they should be separated, so that the xfail is as narrow as possible):

def test_blah_logp(self):

   condition=(aesara.config.floatX == "float32"),
   reason="Fails on float32 due to numerical issues",
def test_blah_logcdf(self):

5. Adding tests for the moment method#

Tests for the moment make use of the function assert_moment_is_expected which checks if:

  1. Moments return the expected values

  2. Moments have the expected size and shape

  3. Moments have a finite logp

import pytest
from pymc.distributions import Blah
from pymc.tests.distributions.util import assert_moment_is_expected

    "param1, param2, size, expected",
        (0, 1, None, 0),
        (0, np.ones(5), None, np.zeros(5)),
        (np.arange(5), 1, None, np.arange(5)),
        (np.arange(5), np.arange(1, 6), (2, 5), np.full((2, 5), np.arange(5))),
def test_blah_moment(param1, param2, size, expected):
    with Model() as model:
        Blah("x", param1=param1, param2=param2, size=size)
    assert_moment_is_expected(model, expected)

Here are some details worth keeping in mind:

  1. In the case where you have to manually broadcast the parameters with each other it’s important to add test conditions that would fail if you were not to do that. A straightforward way to do this is to make the used parameter a scalar, the unused one(s) a vector (one at a time) and size None.

  2. In other words, make sure to test different combinations of size and broadcasting to cover these cases.

6. Documenting the new Distribution#

New distributions should have a rich docstring, following the same format as that of previously implemented distributions. It generally looks something like this:

r"""Univariate blah distribution.

 The pdf of this distribution is

 .. math::

    f(x \mid \param1, \param2) = \exp{x * (param1 + \log{param2})}

 .. plot::

     import matplotlib.pyplot as plt
     import numpy as np
     import scipy.stats as st
     import arviz as az
     x = np.linspace(-5, 5, 1000)
     params1 = [0., 0., 0., -2.]
     params2 = [0.4, 1., 2., 0.4]
     for param1, param2 in zip(params1, params2):
         pdf = st.blah.pdf(x, param1, param2)
         plt.plot(x, pdf, label=r'$\param1$ = {}, $\param2$ = {}'.format(param1, param2))
     plt.xlabel('x', fontsize=12)
     plt.ylabel('f(x)', fontsize=12)

 ========  ==========================================
 Support   :math:`x \in [0, \infty)`
 ========  ==========================================

 Blah distribution can be parameterized either in terms of param2 or
 alt_param2. The link between the two parametrizations is
 given by

 .. math::

    \param2 = \dfrac{1}{\alt_param2}

 param1: float
     Interpretation of param1.
 param2: float
     Interpretation of param2 (param2 > 0).
 alt_param2: float
     Interpretation of alt_param2 (alt_param2 > 0) (alternative to param2).

 .. code-block:: python

     with pm.Model():
         x = pm.Blah('x', param1=0, param2=10)

The new distribution should be referenced in the respective API page in the docs module (e.g., pymc/docs/api/distributions.continuous.rst). If appropriate, a new notebook example should be added to pymc-examples illustrating how this distribution can be used and how it relates (and/or differs) from other distributions that users are more likely to be familiar with.