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+1,129 @@ +# | # Bayesian Linear Regression via BlackJAX Nested Sampling +# | +# | This minimal example demonstrates Bayesian parameter estimation for a +# | simple linear model using adaptive nested sampling. Synthetic data is +# | generated from a linear model with parameters for slope, intercept, and +# | noise # | level. The example sets up the likelihood and uniform priors, +# | samples the posterior using BlackJax’s nested sampling routine, and finally +# | post-processes the samples with the Anesthetic package for in-depth +# | visualization and analysis. +# | +# | ## Installation +# |```bash +# | python -m venv venv +# | source venv/bin/activate +# | pip install git+https://github.com/handley-lab/blackjax@proposal +# | pip install anesthetic tqdm +# | python line.py +# |``` + +import blackjax +import jax +import jax.numpy as jnp +import tqdm + + +jax.config.update('jax_enable_x64', True) + +# | Nested Sampling parameters +rng_key = jax.random.PRNGKey(0) +n_dims = 3 +n_live = 1000 +n_delete = 500 +num_mcmc_steps = n_dims * 5 + +# | Define data and likelihood +x = jnp.linspace(-1, 1, 10) +m = 2.0 +c = 1.0 +sigma = 0.1 +key, rng_key = jax.random.split(rng_key) +y = m * x + c + sigma * jax.random.normal(key, (10,), dtype=jnp.float64) + +# | Plot the data +import matplotlib.pyplot as plt +plt.errorbar(x, y, yerr=sigma, fmt="o", label="data") +plt.plot(x, m * x + c, label="true model") +plt.legend() +plt.show() + +@jax.jit +def loglikelihood_fn(p): + return jax.scipy.stats.multivariate_normal.logpdf(y, p["m"] * x + p["c"], p["sigma"]) + +# | Define the prior function + +m_min, m_max = -10.0, 10.0 +c_min, c_max = -10.0, 10.0 +sigma_min, sigma_max = 0.0, 10.0 + +uniform_logprob = lambda x, a, b: jax.scipy.stats.uniform.logpdf(x, a, b-a) + +def logprior_fn(p): + logprior = 0.0 + logprior += uniform_logprob(p["m"], m_min, m_max) + logprior += uniform_logprob(p["c"], c_min, c_max) + logprior += uniform_logprob(p["sigma"], sigma_min, sigma_max) + return logprior + +# | Sample live points from the prior +rng_key, init_key = jax.random.split(rng_key, 2) +init_keys = jax.random.split(init_key, n_dims) +particles = { + "m": jax.random.uniform(init_keys[0], (n_live,), minval=m_min, maxval=m_max), + "c": jax.random.uniform(init_keys[1], (n_live,), minval=c_min, maxval=c_max), + "sigma": jax.random.uniform(init_keys[2], (n_live,), minval=sigma_min, maxval=sigma_max), + } + +_, ravel_fn = jax.flatten_util.ravel_pytree({k: v[0] for k, v in particles.items()}) + +# | Initialize the Nested Sampling algorithm +nested_sampler = blackjax.ns.adaptive.nss( + logprior_fn=logprior_fn, + loglikelihood_fn=loglikelihood_fn, + n_delete=n_delete, + num_mcmc_steps=num_mcmc_steps, + ravel_fn=ravel_fn, +) + +state = nested_sampler.init(particles, loglikelihood_fn) + +@jax.jit +def one_step(carry, xs): + state, k = carry + k, subk = jax.random.split(k, 2) + state, dead_point = nested_sampler.step(subk, state) + return (state, k), dead_point + +# | Run Nested Sampling +dead = [] +with tqdm.tqdm(desc="Dead points", unit=" dead points") as pbar: + while not state.sampler_state.logZ_live - state.sampler_state.logZ < -3: + (state, rng_key), dead_info = one_step((state, rng_key), None) + dead.append(dead_info) + pbar.update(n_delete) # Update progress bar + +# | anesthetic post-processing +from anesthetic import NestedSamples +import numpy as np +dead = jax.tree.map( + lambda *args: jnp.reshape(jnp.stack(args, axis=0), + (-1,) + args[0].shape[1:]), + *dead) + +live = state.sampler_state +logL = np.concatenate((dead.logL, live.logL), dtype=float) +logL_birth = np.concatenate((dead.logL_birth, live.logL_birth), dtype=float) +data = np.concatenate([ + np.column_stack([v for v in dead.particles.values()]), + np.column_stack([v for v in live.particles.values()]) + ], axis=0) + +columns = list(dead.particles.keys()) +samples = NestedSamples(data, logL=logL, logL_birth=logL_birth, columns=columns) +samples.to_csv('line.csv') + +# | load results from file if quicker +from anesthetic import read_chains +samples = read_chains("line.csv") +samples.gui()