jeffreys_prior_binomial.ipynb

Kernel: Unknown Kernel

# jeffreys prior for bernoulli using 2 paramterizatiobs
# fig 1.9 of 'Bayeysian Modeling and Computation'


import numpy as np
import matplotlib.pyplot as plt

try:
    import probml_utils as pml
except ModuleNotFoundError:
    %pip install -qq git+https://github.com/probml/probml-utils.git
    import probml_utils as pml

θ = np.linspace(0, 1, 100)
κ = θ / (1 - θ)
y = 2
n = 7

_, ax = plt.subplots(2, 2, figsize=(10, 5), sharex="col", sharey="row")

ax[0, 0].set_title("Jeffreys' prior for Alice")
ax[0, 0].plot(θ, θ ** (-0.5) * (1 - θ) ** (-0.5))
ax[1, 0].set_title("Jeffreys' posterior for Alice")
ax[1, 0].plot(θ, θ ** (y - 0.5) * (1 - θ) ** (n - y - 0.5))
ax[1, 0].set_xlabel("θ")
ax[0, 1].set_title("Jeffreys' prior for Bob")
ax[0, 1].plot(κ, κ ** (-0.5) * (1 + κ) ** (-1))
ax[1, 1].set_title("Jeffreys' posterior for Bob")
ax[1, 1].plot(κ, κ ** (y - 0.5) * (1 + κ) ** (-n - 1))
ax[1, 1].set_xlim(-0.5, 10)
ax[1, 1].set_xlabel(r"$\phi$")
ax[1, 1].text(-4.0, 0.030, size=18, s=r"$p(\theta \mid x) \, \frac{d\theta}{d\phi}$")
ax[1, 1].annotate("", xy=(-0.5, 0.025), xytext=(-4.5, 0.025), arrowprops=dict(facecolor="black", shrink=0.05))
ax[1, 1].text(-4.0, 0.007, size=18, s=r"$p(\phi \mid x) \, \frac{d\phi}{d\theta}$")
ax[1, 1].annotate(
    "", xy=(-4.5, 0.015), xytext=(-0.5, 0.015), arrowprops=dict(facecolor="black", shrink=0.05), annotation_clip=False
)

plt.subplots_adjust(wspace=0.4, hspace=0.4)
plt.tight_layout()
pml.savefig("jeffreys_priors.pdf", dpi=300)