Prior elicitation

What is the meaning of the parameters in the various processes? Let’s study the prior distribution of the number of clusters!

import numpy as np
import matplotlib.pyplot as plt

from pybmix.core.mixing import DirichletProcessMixing, PitmanYorMixing, StickBreakMixing

Dirichlet Process

Compare how the prior distribution varies for a DP for different values of the ‘total_mass’ parameter

nsamples = 100
grid = np.arange(1, 30)

for total_mass in [0.25, 1, 2, 3, 5, 7]:
    dp = DirichletProcessMixing(total_mass=total_mass)
    probs = dp.prior_cluster_distribution(grid, nsamples)
    plt.plot(grid, probs, label="total_mass={0}".format(total_mass))

plt.legend()
plt.show()

Pitman-Yor Process

The PY process is controlled by two parameters: ‘strength’ (plays the same role of ‘total_mass’ in the Dirichlet Process) and ‘discount’. For a fixed value of ‘strenght’, see how the distribution varies for different values of ‘discount’.

strength = 1
for discount in [0.001, 0.1, 0.2, 0.5]:
    py = PitmanYorMixing(strength=strength, discount=discount)
    probs = py.prior_cluster_distribution(grid, nsamples)
    plt.plot(grid, probs, label="strength={0}, discount={1}".format(strength, discount))

plt.legend()
plt.show()

Stick Breaking prior

Truncation of a Dirichlet Process

n = [50, 1000]
H = [10, 20, 50]
grid = np.arange(1, 50)

fig, axes = plt.subplots(nrows=len(n), ncols=len(H), figsize=(15, 10))
for i, nval in enumerate(n):
    for j, hval in enumerate(H):
        for total_mass in [0.25, 1, 2, 3, 5, 7]:
            sb = StickBreakMixing(n_comp=hval, strength=total_mass)
            probs = sb.prior_cluster_distribution(grid, nval, mc_iter=10000)
            axes[i][j].plot(grid, probs, label="total_mass={0}".format(total_mass))
            axes[i][j].set_title("N: {0}, H: {1}".format(nval, hval), fontsize=18)

axes[1, 1].legend(ncol=3, fontsize=14, bbox_to_anchor=(1.5, -0.1))
plt.show()

Truncation of a Pitman-Yor Process

n = [200, 1000]
H = [10, 20, 50]
strength = 1
discount = np.linspace(0.2, 0.9, 4)

fig, axes = plt.subplots(nrows=len(n), ncols=len(H), figsize=(15, 10))

for i, nval in enumerate(n):
    for j, hval in enumerate(H):
        for d in discount:
            sb = StickBreakMixing(n_comp=hval, strength=strength, discount=d)
            probs = sb.prior_cluster_distribution(grid, nval, mc_iter=10000)
            axes[i][j].plot(grid, probs, label="discount={0:.2f}".format(d))
            axes[i][j].set_title("N: {0}, H: {1}".format(nval, hval), fontsize=18)

axes[1, 1].legend(ncol=2, fontsize=14, bbox_to_anchor=(1.2, -0.1))
plt.show()