Add example of a discrete-continuous mixture

workflows/Mixtures/figure_discrete_continuous.smk

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+"""Distribution with continuous two-dimensional X and binary Y."""
+import json
+from jax import config
+config.update("jax_enable_x64", True)
+
+import jax
+import jax.numpy as jnp
+
+import matplotlib
+import matplotlib.pyplot as plt
+matplotlib.use('Agg')
+import numpy as np
+
+import bmi
+from bmi.samplers import fine
+
+from tensorflow_probability.substrates import jax as tfp
+tfd = tfp.distributions
+
+
+# === WORKDIR ===
+workdir: "generated/mixtures/discrete_continuous/"
+
+
+def construct_bernoulli(p: float, dtype=jnp.float64) -> tfd.Distribution:
+    """Constructs a Bernoulli distribution, as
+    TensorFlow Probability disallows products of continuous
+    and discrete distributions."""
+    return tfd.Independent(
+            tfd.Bernoulli(probs=jnp.asarray([p], dtype=dtype), dtype=dtype),
+            reinterpreted_batch_ndims=1,
+    )
+
+
+def define_distribution():
+    """Defines the joint distribution P(X, Y)."""
+    # Define the distributions for X_k
+    x1 = fine.construct_multivariate_student_distribution(
+        mean=-jnp.ones(2),
+        dispersion=0.2 * jnp.eye(2),
+        df=8,
+    )
+    x2 = fine.construct_multivariate_normal_distribution(
+        mean=jnp.zeros(2),
+        covariance=0.1 * bmi.samplers.canonical_correlation([0.95]),
+    )
+    x3 = fine.construct_multivariate_student_distribution(
+        mean=jnp.ones(2),
+        dispersion=0.2 * jnp.eye(2),
+        df=5,
+    )
+
+    # Define the distributions for Y_k
+    y1 = construct_bernoulli(0.95)
+    y2 = construct_bernoulli(0.5)
+    y3 = construct_bernoulli(0.05)
+
+    # Define the mixture distribution
+    components = [
+        fine.ProductDistribution(dist_x, dist_y)
+        for dist_x, dist_y in zip([x1, x2, x3], [y1, y2, y3])
+    ]
+    connect_prob = 0.5
+    bulk_prob = 0.5 * (1 - connect_prob)
+    joint_distribution = fine.mixture(proportions=[bulk_prob, connect_prob, bulk_prob], components=components)
+
+    return joint_distribution
+
+rule all:
+    input:
+        "estimate.json",
+        "figure_discrete_continuous_example.pdf"
+
+
+rule estimate_mi:
+    output: "estimate.json"
+    run:
+        key = jax.random.PRNGKey(121)
+        mi, mi_std_err = fine.monte_carlo_mi_estimate(key, dist=define_distribution(), n=1_000_000)
+        with open(output[0], "w") as fp:
+            json.dump({"estimate": float(mi), "std_err": float(mi_std_err)}, fp=fp)
+
+rule sample_joint:
+    output: "joint_samples.npz"
+    run:
+        key = jax.random.PRNGKey(1000)
+        x, y = define_distribution().sample(1000, key)
+        np.savez(output[0], x=x, y=y)
+
+rule sample_pmi:
+    output: "pmi_samples.npz"
+    run:
+        key = jax.random.PRNGKey(101)
+        profile_samples = fine.pmi_profile(key=key, dist=define_distribution(), n=1_000_000)
+        np.savez(output[0], samples=profile_samples)
+
+
+rule plot_figure:
+    input:
+        joint_samples="joint_samples.npz",
+        pmi_samples="pmi_samples.npz"
+    output:
+        figure = "figure_discrete_continuous_example.pdf"
+    run:
+        samples = np.load(input.joint_samples)
+        xs, ys = samples["x"], samples["y"]
+        pmis = np.load(input.pmi_samples)["samples"]
+
+        fig, axs = plt.subplots(1, 2, figsize=(5, 2.5), dpi=300)
+
+        ax = axs[0]
+        colors = ['blue' if y < 0.5 else 'orange' for y in ys.ravel()]
+        ax.scatter(xs[..., 0], xs[..., 1], c=colors, s=1, alpha=0.8, rasterized=True, marker=".")
+        ax.set_title("Samples from $P_{XY}$")
+        ax.set_xlabel("$x_1$")
+        ax.set_ylabel("$x_2$")
+
+        ax = axs[1]
+        ax.hist(pmis, bins=100, density=True, color='black', alpha=0.7, rasterized=True)
+        ax.set_xlim(-3, 1)
+        ax.set_title("PMI profile")
+        ax.set_xlabel("PMI")
+        ax.set_ylabel("Density")    
+
+        for ax in axs:
+            ax.spines[['top', 'right']].set_visible(False)
+
+        fig.tight_layout()
+        fig.savefig(output.figure)