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Exercise - Inverse Transform Sampling

Table of Contents

Introduction

Inverse sampling is a usefull method to sample from a distribution$ p(x) $, where it is hard to sample from but easy to find and invert the$ cdf $ (cumulative distribution function). Having the inverse of the$ cdf $ we just need to draw a sample from$ \mathcal U (0,1) $ and act with the inverse$ cdf $ on it to get our desired sample for$ p(x) $. In this notebook you will implement this for the sample example of a linear distribution function.

Requirements

Knowledge

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

  • Monte Carlo estimator
  • Inverse transform sampling

The following material can help you to acquire this knowledge:

Python Modules

# External Modules
import matplotlib.pyplot as plt
import numpy as np
from numpy.testing import assert_almost_equal

%matplotlib inline

Exercise

Task:

Write a function, which estimates$ \mathbb E_{x \sim p} [f(x)] $ with the use of$ n $ drawn samples, with the given functions:

$ f(x)=x^2 $

and

\begin{equation} p(x) = \begin{cases} \frac{1}{2}x & \text{for } x \in [0, 2] \\ 0 & \text{for } x \notin [0, 2] \\ \end{cases} \end{equation}

Hint:

To complete this task, do the following for all$ i \in \{1, 2, \dots, n \} $:

  1. Draw one$ x_i $ from$ p(x) $ using inverse transform sampling, also called Smirnov Transform

  2. Calculate$ f_i = f(x_i) $

Then Calculate the mean value of all$ f_i $s

# Complete this function

x = np.linspace(0,2, int(10e5))

np.random.seed(42) # for reproducable results

def sample_x(n=1):
    """
    Samples n values from the pdf p(x)=Cx 
    using inverse transform sampling.
    
    :param n: The number of samples to draw and return
    :type n: int

    :returns: Sampled xs as 1D numpy.ndarray,
            uniformly distributed ys used to sample xs
    :rtype: numpy.ndarray[float], 
            numpy.ndarray[float]
    """
    raise NotImplementedError()

x_, y_ = sample_x(50000)
plt.hist(x_, density=True)
plt.plot(x,0.5*x)
plt.xlabel("x")
_ = plt.ylabel("p(x)")
# Complete this function. Use your implemented sample_x function inside.

f = lambda x:x**2 
def estimate_f(n=100, f=f):
    """
    Estimates the expected value of function f with respect
    to p(x)=Cx using inverse transform sampling.
    
    :param n: The number of samples to draw and return
    :type n: int
    :param f: The function to estimate
    :type f: lambda-function

    :returns: the estimated expected value as mean,
            a list of the single results of f from the sampled xs,
            x values of the drawn samples,
            y values used to draw the samples
    :rtype: float,
            List[float],
            numpy.ndarray[float], 
            numpy.ndarray[float]
    """
    raise NotImplementedError()

fs_mean, fs, x_, y_ = estimate_f(1000,f)
print(f_mean)
# Executing this cell must not throw an Exception
# The solution is obfuscated, so you can solve the exercise without unintendedly spoiling yourself

obfuscated_solution = 24.911757965 / 54345 * 4363
assert_almost_equal(fs_mean,obfuscated_solution, decimal=1)

Literature

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 - Inverse Transform 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.