student_laplace_pdf_plot.ipynb

Kernel: Python [conda env:probml_murphy]

Univariate Gaussian, Laplace and Student t-distribution

import jax
import jax.numpy as jnp
import seaborn as sns
import matplotlib.pyplot as plt

try:
    from probml_utils import latexify, savefig, is_latexify_enabled
except ModuleNotFoundError:
    %pip install -qq git+https://github.com/probml/probml-utils.git
    from probml_utils import latexify, savefig, is_latexify_enabled

from jax.scipy.stats import t, norm, laplace

latexify(fig_width=0.45 * 6, fig_height=1.5)

/home/patel_zeel/miniconda3/envs/probml_murphy/lib/python3.9/site-packages/probml_utils/plotting.py:26: UserWarning: LATEXIFY environment variable not set, not latexifying
  warnings.warn("LATEXIFY environment variable not set, not latexifying")

x = jnp.linspace(-4, 4, 100)
normal = norm.pdf(x, loc=0, scale=1)
laplace_ = laplace.pdf(x, loc=0, scale=1 / (2**0.5))
student_t1 = t.pdf(x, df=1, loc=0, scale=1)
student_t2 = t.pdf(x, df=2, loc=0, scale=1)

WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

LEGEND_SIZE = 7 if is_latexify_enabled() else None

plt.figure()

ax = plt.gca()
(t1_plot,) = plt.plot(x, student_t1, "b--", label="Student\n" + r"$(\nu=1)$")
(t2_plot,) = plt.plot(x, student_t2, "g--", label="Student\n" + r"$(\nu=2)$")
(norm_plot,) = plt.plot(x, normal, "k:", label="Gaussian")
(laplace_plot,) = plt.plot(x, laplace_, "r-", label="Laplace")

legend1 = plt.legend(handles=[t1_plot, norm_plot], loc="upper right", prop={"size": LEGEND_SIZE})
ax.add_artist(legend1)
legend2 = plt.legend(handles=[t2_plot, laplace_plot], loc="upper left", prop={"size": LEGEND_SIZE})
ax.add_artist(legend2)

plt.ylabel("pdf")
plt.xlabel("$x$")
sns.despine()
savefig("studentLaplacePdf2.pdf")

/home/patel_zeel/miniconda3/envs/probml_murphy/lib/python3.9/site-packages/probml_utils/plotting.py:79: UserWarning: set FIG_DIR environment variable to save figures
  warnings.warn("set FIG_DIR environment variable to save figures")

plt.figure()
plt.plot(
    x, jnp.log(normal), "k:", x, jnp.log(student_t1), "b--", x, jnp.log(student_t2), "g--", x, jnp.log(laplace_), "r-"
)


plt.ylabel("log pdf")
plt.xlabel("$x$")
plt.legend(("Gaussian", "Student " + r"$(\nu=1)$", "Student " + r"$(\nu=2)$", "Laplace"), prop={"size": LEGEND_SIZE})
sns.despine()
savefig("studentLaplaceLogpdf2.pdf")

/home/patel_zeel/miniconda3/envs/probml_murphy/lib/python3.9/site-packages/probml_utils/plotting.py:79: UserWarning: set FIG_DIR environment variable to save figures
  warnings.warn("set FIG_DIR environment variable to save figures")

Demo

You can visualize the effect of loc, scale, and degree of freedom on the above three distributions by changing them.

def make_graph(data):
    plt.figure(figsize=(8, 6))
    x = jnp.linspace(-4, 4, 100)
    deg_of_freedom = data["dof"]
    loc = data["loc"]
    scale = data["scale"]

    student_t = t.pdf(x, df=deg_of_freedom, loc=loc, scale=scale)
    normal = norm.pdf(x, loc=loc, scale=scale)
    laplace_ = laplace.pdf(x, loc=loc, scale=scale)

    plt.plot(x, normal, "k:", x, student_t, "g--", x, laplace_, "r-")
    plt.ylim(0, 0.6)
    plt.legend(("Gaussian", "Student", "Laplace"), bbox_to_anchor=(1.1, 1))
    sns.despine()

from ipywidgets import interact


@interact(dof=(0.1, 10), loc=(-1.1, 1.1), scale=(0.7, 2))
def generate_random(dof, loc, scale):
    data = {}
    data["dof"] = dof
    data["loc"] = loc
    data["scale"] = scale

    make_graph(data);

interactive(children=(FloatSlider(value=5.05, description='dof', max=10.0, min=0.1), FloatSlider(value=0.0, de…