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Generalized Numerical Inputs

Conjugate models work with anything that works like numbers.

Here are examples of the Binomial and Beta distributions with different packages data as input. For more details on this model, see the Binomial Model example.

Setup

Import the binomial_beta model and the Beta distribution:

from conjugate.distributions import Beta
from conjugate.models import binomial_beta

Polars

Bayesian models with the Polars package:

import polars as pl

# Data
df = pl.DataFrame({"total": [10, 20, 50], "successes": [5, 10, 25]})

# Conjugate prior
prior = Beta(alpha=1, beta=1)
posterior = binomial_beta(n=df["total"], x=df["successes"], prior=prior)

ax = posterior.plot_pdf(label=df["total"])
ax.legend(title="sample size")

Polars Example

Narwhals

The Narwhals package allows users to bring their own DataFrame library.

Their API is subset of polars:

import narwhals as nw


@nw.narwhalify
def bayesian_inference(df, prior):
    N = nw.col("total")
    X = nw.col("successes")

    posterior = binomial_beta(n=N, x=X, prior=prior)

    return df.with_columns(
        posterior_alpha=posterior.alpha, posterior_beta=posterior.beta
    )

But works for various DataFrame libraries! Try Polars

df = pl.DataFrame({"total": [10, 20, 50], "successes": [5, 10, 25]})

prior = Beta(alpha=1, beta=1)
bayesian_inference(df, prior)

shape: (3, 4)
┌───────┬───────────┬─────────────────┬────────────────┐
│ total ┆ successes ┆ posterior_alpha ┆ posterior_beta │
│ ---   ┆ ---       ┆ ---             ┆ ---            │
│ i64   ┆ i64       ┆ i64             ┆ i64            │
╞═══════╪═══════════╪═════════════════╪════════════════╡
│ 10    ┆ 5         ┆ 6               ┆ 6              │
│ 20    ┆ 10        ┆ 11              ┆ 11             │
│ 50    ┆ 25        ┆ 26              ┆ 26             │
└───────┴───────────┴─────────────────┴────────────────┘

Or Pandas

df = pd.DataFrame({"total": [10, 20, 50], "successes": [5, 10, 25]})

prior = Beta(alpha=1, beta=1)
bayesian_inference(df, prior)

   total  successes  posterior_alpha  posterior_beta
0     10          5                6               6
1     20         10               11              11
2     50         25               26              26

Models with SQL

Bayesian models in SQL using the SQL Builder, PyPika:

from pypika import Field

# Columns from table in database
N = Field("total")
X = Field("successes")

# Conjugate prior
prior = Beta(alpha=1, beta=1)
posterior = binomial_beta(n=N, x=X, prior=prior)

print("Posterior alpha:", posterior.alpha)
print("Posterior beta:", posterior.beta)
# Posterior alpha: 1+"successes"
# Posterior beta: 1+"total"-"successes"

Even the priors can be fields too:

alpha = Field("previous_successes") - 1
beta = Field("previous_failures") - 1

prior = Beta(alpha=alpha, beta=beta)
posterior = binomial_beta(n=N, x=X, prior=prior)

print("Posterior alpha:", posterior.alpha)
print("Posterior beta:", posterior.beta)
# Posterior alpha: "previous_successes"-1+"successes"
# Posterior beta: "previous_failures"-1+"total"-"successes"

PyMC

Use PyMC distributions for sampling with additional uncertainty:

import pymc as pm

alpha = pm.Gamma.dist(alpha=1, beta=20)
beta = pm.Gamma.dist(alpha=1, beta=20)

# Observed Data
N = 10
X = 4

# Conjugate prior
prior = Beta(alpha=alpha, beta=beta)
posterior = binomial_beta(n=N, x=X, prior=prior)

# Reconstruct the posterior distribution with PyMC
prior_dist = pm.Beta.dist(alpha=prior.alpha, beta=prior.beta)
posterior_dist = pm.Beta.dist(alpha=posterior.alpha, beta=posterior.beta)

samples = pm.draw([alpha, beta, prior_dist, posterior_dist], draws=1000)

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