numpyro

What is NumPyro?

NumPyro is a lightweight probabilistic programming library that provides a NumPy backend for Pyro. We rely on [JAX][JAX] for automatic differentiation and JIT compilation to GPU / CPU. NumPyro is under active development, so beware of brittleness, bugs, and changes to the API as the design evolves.

NumPyro is designed to be lightweight and focuses on providing a flexible substrate that users can build on:

• Pyro Primitives: NumPyro programs can contain regular Python and NumPy code, in addition to Pyro primitives like sample and param. The model code should look very similar to Pyro except for some minor differences between PyTorch and Numpy's API. See the example below.
• Inference algorithms: NumPyro supports a number of inference algorithms, with a particular focus on MCMC algorithms like Hamiltonian Monte Carlo, including an implementation of the No U-Turn Sampler. Additional MCMC algorithms include MixedHMC (which can accommodate discrete latent variables) as well as HMCECS (which only computes the likelihood for subsets of the data in each iteration). One of the motivations for NumPyro was to speed up Hamiltonian Monte Carlo by JIT compiling the verlet integrator that includes multiple gradient computations. With JAX, we can compose jit and grad to compile the entire integration step into an XLA optimized kernel. We also eliminate Python overhead by JIT compiling the entire tree building stage in NUTS (this is possible using Iterative NUTS). There is also a basic Variational Inference implementation together with many flexible (auto)guides for Automatic Differentiation Variational Inference (ADVI). The variational inference implementation supports a number of features, including support for models with discrete latent variables (see TraceGraph_ELBO and TraceEnum_ELBO).
• Distributions: The numpyro.distributions module provides distribution classes, constraints and bijective transforms. The distribution classes wrap over samplers implemented to work with JAX's functional pseudo-random number generator. The design of the distributions module largely follows from PyTorch. A major subset of the API is implemented, and it contains most of the common distributions that exist in PyTorch. As a result, Pyro and PyTorch users can rely on the same API and batching semantics as in torch.distributions. In addition to distributions, constraints and transforms are very useful when operating on distribution classes with bounded support. Finally, distributions from TensorFlow Probability (TFP) can directly be used in NumPyro models.
• Effect handlers: Like Pyro, primitives like sample and param can be provided nonstandard interpretations using effect-handlers from the numpyro.handlers module, and these can be easily extended to implement custom inference algorithms and inference utilities.

A Simple Example - 8 Schools

Let us explore NumPyro using a simple example. We will use the eight schools example from Gelman et al., Bayesian Data Analysis: Sec. 5.5, 2003, which studies the effect of coaching on SAT performance in eight schools.

The data is given by:

>>> import numpy as np

>>> J = 8
>>> y = np.array([28.0, 8.0, -3.0, 7.0, -1.0, 1.0, 18.0, 12.0])
>>> sigma = np.array([15.0, 10.0, 16.0, 11.0, 9.0, 11.0, 10.0, 18.0])

, where y are the treatment effects and sigma the standard error. We build a hierarchical model for the study where we assume that the group-level parameters theta for each school are sampled from a Normal distribution with unknown mean mu and standard deviation tau, while the observed data are in turn generated from a Normal distribution with mean and standard deviation given by theta (true effect) and sigma, respectively. This allows us to estimate the population-level parameters mu and tau by pooling from all the observations, while still allowing for individual variation amongst the schools using the group-level theta parameters.

>>> import numpyro
>>> import numpyro.distributions as dist

>>> # Eight Schools example
... def eight_schools(J, sigma, y=None):
...     mu = numpyro.sample('mu', dist.Normal(0, 5))
...     tau = numpyro.sample('tau', dist.HalfCauchy(5))
...     with numpyro.plate('J', J):
...         theta = numpyro.sample('theta', dist.Normal(mu, tau))
...         numpyro.sample('obs', dist.Normal(theta, sigma), obs=y)

Let us infer the values of the unknown parameters in our model by running MCMC using the No-U-Turn Sampler (NUTS). Note the usage of the extra_fields argument in MCMC.run. By default, we only collect samples from the target (posterior) distribution when we run inference using MCMC. However, collecting additional fields like potential energy or the acceptance probability of a sample can be easily achieved by using the extra_fields argument. For a list of possible fields that can be collected, see the HMCState object. In this example, we will additionally collect the potential_energy for each sample.

>>> from jax import random
>>> from numpyro.infer import MCMC, NUTS

>>> nuts_kernel = NUTS(eight_schools)
>>> mcmc = MCMC(nuts_kernel, num_warmup=500, num_samples=1000)
>>> rng_key = random.key(0)
>>> mcmc.run(rng_key, J, sigma, y=y, extra_fields=('potential_energy',))

We can print the summary of the MCMC run, and examine if we observed any divergences during inference. Additionally, since we collected the potential energy for each of the samples, we can easily compute the expected log joint density.

…

The values above 1 for the split Gelman Rubin diagnostic (r_hat) indicates that the chain has not fully converged. The low value for the effective sample size (n_eff), particularly for tau, and the number of divergent transitions looks problematic. Fortunately, this is a common pathology that can be rectified by using a non-centered parameterization for tau in our model. This is straightforward to do in NumPyro by using a TransformedDistribution instance together with a reparameterization effect handler. Let us rewrite the same model but instead of sampling theta from a Normal(mu, tau), we will instead sample it from a base Normal(0, 1) distribution that is transformed using an AffineTransform. Note that by doing so, NumPyro runs HMC by generating samples theta_base for the base Normal(0, 1) distribution instead. We see that the resulting chain does not suffer from the same pathology — the Gelman Rubin diagnostic is 1 for all the parameters and the effective sample size looks quite good!

…

Note that for the class of distributions with loc,scale parameters such as Normal, Cauchy, StudentT, we also provide a LocScaleReparam reparameterizer to achieve the same purpose. The corresponding code will be

with numpyro.handlers.reparam(config={'theta': LocScaleReparam(centered=0)}):
    theta = numpyro.sample('theta', dist.Normal(mu, tau))

Now, let us assume that we have a new school for which we have not observed any test scores, but we would like to generate predictions. NumPyro provides a Predictive class for such a purpose. Note that in the absence of any observed data, we simply use the population-level parameters to generate predictions. The Predictive utility conditions the unobserved mu and tau sites to values drawn from the posterior distribution from our last MCMC run, and runs the model forward to generate predictions.

>>> from numpyro.infer import Predictive

>>> # New School
... def new_school():
...     mu = numpyro.sample('mu', dist.Normal(0, 5))
...     tau = numpyro.sample('tau', dist.HalfCauchy(5))
...     return numpyro.sample('obs', dist.Normal(mu, tau))

>>> predictive = Predictive(new_school, mcmc.get_samples())
>>> samples_predictive = predictive(random.key(1))
>>> print(np.mean(samples_predictive['obs']))  # doctest: +SKIP
3.9886456

More Examples

For some more examples on specifying models and doing inference in NumPyro:

• Bayesian Regression in NumPyro - Start here to get acquainted with writing a simple model in NumPyro, MCMC inference API, effect handlers and writing custom inference utilities.
• Time Series Forecasting - Illustrates how to convert for loops in the model to JAX's lax.scan primitive for fast inference.
• Annotation examples - Illustrates how to utilize the enumeration mechanism to perform inference for models with discrete latent variables.
• Baseball example - Using NUTS for a simple hierarchical model. Compare this with the baseball example in Pyro.
• Hidden Markov Model in NumPyro as compared to Stan.
• Variational Autoencoder - As a simple example that uses Variational Inference with neural networks. Pyro implementation for comparison.
• Gaussian Process - Provides a simple example to use NUTS to sample from the posterior over the hyper-parameters of a Gaussian Process.
• Horseshoe Regression - Shows how to implement generalized linear models equipped with a Horseshoe prior for both binary-valued and real-valued outputs.
• Statistical Rethinking with NumPyro - Notebooks containing translation of the code in Richard McElreath's Statistical Rethinking book second version, to NumPyro.
• Other model examples can be found in the examples site.

Pyro users will note that the API for model specification and inference is largely the same as Pyro, including the distributions API, by design. However, there are some important core differences (r