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Exercise - Rejection Sampling

Table of Contents

Introduction

Rejection sampling is a simple and straight forward algorihtm to generate samples for distributions, which a hard or impossible to sample from, using a second enclosing distribtuion. Using the enclosing function we will sample points and accept them as sample points for our desired distribution if they lie under the desired distribution curve and otherwise reject them.

Requirements

Knowledge

To complete this exercise notebook, you should possess knowledge about the following topics.

  • Rejection Sampling

The following material can help you to acquire this knowledge:

Python Modules

import numpy as np
from matplotlib import pyplot as plt
from scipy.stats import norm

%matplotlib inline

Exercise

We want to sample from the green distribution$ p(x) $ below.

But for now we assume that we can't sample from$ p(x) $ directly, but we can sample from the red one$ q(x) $, and we can compute$ p(x) $ at each position$ x $. 

size=40
sigma_square_1 = 4.0
sigma_square_2 = 4.0
mu_1, sigma_1 = -2.5,np.sqrt(sigma_square_1)
mu_2, sigma_2 = 3.5,np.sqrt(sigma_square_1)
prob_1 = 0.4

rv_1 = norm(loc = mu_1, scale = sigma_1)
rv_2 = norm(loc = mu_2, scale = sigma_2)
x_ = np.arange(-14, 16, .1)

p_green = lambda x: prob_1 * rv_1.pdf(x) + (1-prob_1) * rv_2.pdf(x)
plt.plot(x_, p_green(x_) , "g-",label='$p(x)$')

sigma_red,mu_red = 5. , 1.
q_red = norm(loc = mu_red, scale = sigma_red)

plt.plot(x_, q_red.pdf(x_) , "r-",label='$q(x)$')
plt.legend()

_ = plt.xlabel("x")

Task

Implement rejection sampling to get a sample from$ p(x) $ by

  • sampling from$ q(x) $ and
  • rejecting or accept some of the samples
  • visualize the sample.

Hint

Since we need an enclosing function you will need to scale$ q(x) $.

# x-values of red points are rejected
# x-values of green point are accepted
plt.plot(x_, p_green(x_) , "g-")
plt.plot(x_, f_red(x_)  , "r-")
plt.plot(samples, y_accept, 'g.', label='Samples')
plt.plot(rejected_samples, y_reject, 'r.', label='Samples')
_ = plt.hist(samples, bins=50, density=True)
plt.plot(x_, p_green(x_) , "g-")

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise - Rejection Sampling
by Christian Herta, Klaus Strohmenger
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Copyright 2018 Christian Herta, Klaus Strohmenger

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.