blayers 0.2.0

Bayesian layers for NumPyro and Jax

BLayers

The missing layers package for Bayesian inference.

Write code immediately

pip install blayers

deps are: numpyro, jax, and optax.

Concept

Easily build Bayesian models from parts, abstract away the boilerplate, and tweak priors as you wish. Inspiration from Keras and Tensorflow Probability, but made specifically for Numpyro + Jax.

Fit models either using Variational Inference (VI) or your sampling method of choice. Use BLayer's ELBO implementation to do either batched VI or sampling without having to rewrite models.

BLayers helps you write pure Numpyro, so you can integrate it with any Numpyro code to build models of arbitrary complexity. It also gives you a recipe to build more complex layers as you wish.

The starting point

The simplest non-trivial (and most important!) Bayesian regression model form is the adaptive prior,

lmbda ~ HalfNormal(1)
beta  ~ Normal(0, lmbda)
y     ~ Normal(beta * x, 1)

BLayers takes this as its starting point and most fundamental building block, providing the flexible AdaptiveLayer.

from blayers.layers import AdaptiveLayer
from blayers.links import gaussian_link_exp
def model(x, y):
    mu = AdaptiveLayer()('mu', x)
    return gaussian_link_exp(mu, y)

Pure numpyro

All BLayers is doing is writing Numpyro for you under the hood. This model is exacatly equivalent to writing the following, just using way less code.

from numpyro import distributions, sample

def model(x, y):
    # Adaptive layer does all of this
    input_shape = x.shape[1]
    # adaptive prior
    lmbda = sample(
        name="lmbda",
        fn=distributions.HalfNormal(1.),
    )
    # beta coefficients for regression
    beta = sample(
        name="beta",
        fn=distributions.Normal(loc=0., scale=lmbda),
        sample_shape=(input_shape,),
    )
    mu = jnp.einsum('ij,j->i', x, beta)

    # the link function does this
    sigma = sample(name='sigma', fn=distributions.Exponential(1.))
    return sample('obs', distributions.Normal(mu, sigma), obs=y)

Reparameterizing

To fit MCMC models well it is crucial to reparamterize. BLayers helps you do this, automatically reparameterizing the following distributions which Numpyro refers to as LocScale distributions.

LocScaleDist = (
    dist.Normal
    | dist.LogNormal
    | dist.StudentT
    | dist.Cauchy
    | dist.Laplace
    | dist.Gumbel
)

Then, reparam these distributions automatically and fit with Numpyro's built in MCMC methods.

from blayers.layers import AdaptiveLayer
from blayers.links import gaussian_link_exp
from blayers.sampling import autoreparam

data = {...}

@autoreparam
def model(x, y):
    mu = AdaptiveLayer()('mu', x)
    return gaussian_link_exp(mu, y)

kernel = NUTS(model)
mcmc = MCMC(
    kernel,
    num_warmup=500,
    num_samples=1000,
    num_chains=1,
    progress_bar=True,
)
    mcmc.run(
        rng_key,
        **data,
    )

Mixing it up

The AdaptiveLayer is also fully parameterizable via arguments to the class, so let's say you wanted to change the model from

lmbda ~ HalfNormal(1)
beta  ~ Normal(0, lmbda)
y     ~ Normal(beta * x, 1)

to

lmbda ~ Exponential(1.)
beta  ~ LogNormal(0, lmbda)
y     ~ Normal(beta * x, 1)

you can just do this directly via arguments

from numpyro import distributions,
from blayers.layers import AdaptiveLayer
from blayers.links import gaussian_link_exp
def model(x, y):
    mu = AdaptiveLayer(
        lmbda_dist=distributions.Exponential,
        prior_dist=distributions.LogNormal,
        lmbda_kwargs={'rate': 1.},
        prior_kwargs={'loc': 0.}
    )('mu', x)
    return gaussian_link_exp(mu, y)

"Factories"

Since Numpyro traces sample sites and doesn't record any paramters on the class, you can re-use with a particular generative model structure freely.

from numpyro import distributions
from blayers.layers import AdaptiveLayer
from blayers.links import gaussian_link_exp

my_lognormal_layer = AdaptiveLayer(
    lmbda_dist=distributions.Exponential,
    prior_dist=distributions.LogNormal,
    lmbda_kwargs={'rate': 1.},
    prior_kwargs={'loc': 0.}
)

def model(x, y):
    mu = my_lognormal_layer('mu1', x) + my_lognormal_layer('mu2', x**2)
    return gaussian_link_exp(mu, y)

Additional layers

Fixed prior layers

For you purists out there, we also provide a FixedPriorLayer for standard L1/L2 regression.

from blayers.layers import FixedPriorLayer
from blayers.links import gaussian_link_exp
def model(x, y):
    mu = FixedPriorLayer()('mu', x)
    return gaussian_link_exp(mu, y)

Very useful when you have an informative prior.

Bayesian embeddings

We'll keep track of your lookup table for you.

from blayers.layers import EmbeddingLayer
from blayers.links import gaussian_link_exp
EMB_DIM = 8
def model(x, y, x_cat):
    mu = EmbeddingLayer()('mu', x, x_cats, embedding_dim=EMB_DIM)
    return gaussian_link_exp(mu, y)

Old school random effects

A special case of the embedding layer, where EMB_DIM = 1, useful for super fast one-hot encodings (aka random effects)

from blayers.layers import RandomEffectsLayer
from blayers.links import gaussian_link_exp
def model(x, y, x_cat):
    mu = RandomEffectsLayer()('mu', x, x_cats)
    return gaussian_link_exp(mu, y)

Factorization machines

Developed in Rendle 2010 and Rendle 2011, FMs provide a low-rank approximation to the x-by-x interaction matrix. For those familiar with R syntax, it is an approximation to y ~ x:x, excluding the x^2 terms.

To fit the equivalent of an r model like y ~ x*x (all main effects, x^2 terms, and one-way interaction effects), you'd do

from blayers.layers import AdaptiveLayer, FMLayer
from blayers.links import gaussian_link_exp
def model(x, y):
    mu = (
        AdaptiveLayer('x', x) +
        AdaptiveLayer('x2', x**2) +
        FMLayer(low_rank_dim=3)('xx', x)
    )
    return gaussian_link_exp(mu, y)

UV decomp

We also provide a standard UV deccomp for low rank interaction terms

from blayers.layers import AdaptiveLayer, LowRankInteractionLayer
from blayers.links import gaussian_link_exp
def model(x, z, y):
    mu = (
        AdaptiveLayer('x', x) +
        AdaptiveLayer('z', z) +
        LowRankInteractionLayer(low_rank_dim=3)('xz', x, z)
    )
    return gaussian_link_exp(mu, y)

Links

We provide link functions as a convenience to abstract away a bit more Numpyro boilerplate. Link functions take model predictions as inputs to a distribution.

The simplest example is the Gaussian link

mu = ...
sigma ~ Exp(1)
y     ~ Normal(mu, sigma)

We currently provide

• negative_binomial_link
• logit_link
• poission_link
• gaussian_link_exp
• lognormal_link_exp

Link functions include trainable scale parameters when needed, as in the case of Gaussians. We also provide classes for eaisly making additional links via the LocScaleLink and SingleParamLink classes.

For instance, the Poisson link is created like this:

poission_link = SingleParamLink(obs_dist=dists.Poisson)

And implements

rate = ...
y    ~ Poisson(rate)

In a Numpyro model, you use a link like

from blayers.layers import AdaptiveLayer
from blayers.links import poisson_link
def model(x, y):
    rate = AdaptiveLayer()('rate', x)
    return poisson_link(rate, y)

Batched loss

The default Numpyro way to fit batched VI models is to use plate, which confuses me a lot. Instead, BLayers provides Batched_Trace_ELBO which does not require you to use plate to batch in VI. Just drop your model in.

from blayers.infer import Batched_Trace_ELBO, svi_run_batched

svi = SVI(model_fn, guide, optax.adam(schedule), loss=loss_instance)

svi_result = svi_run_batched(
    svi,
    rng_key,
    num_steps,
    batch_size=1000,
    **model_data,
)