Show code cell source Hide code cell source

import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats('svg')
import seaborn as sns
sns.set_context("paper")
sns.set_style("ticks");

Sampling Estimates of the Probability Density via Histograms

As in the previous section, consider a random variable \(X\) and a function of \(X\), \(Y=g(X)\). We wish to approximate the probability density \(p(y)\) of \(Y=g(X)\) from samples. We start by splitting the domain of \(y\) into \(M\) small bins. Assume these bins have bounds \(b_0, b_1, \dots, b_M\). That is, the first bin is \([b_0,b_1]\), the second one is \([b_1,b_2]\), etc. We will approximate \(p(y)\) with a constant inside each bin. That is, the approximation is:

\[ \hat{p}_M(y) = \sum_{j=1}^Mc_j 1_{[b_{j-1}, b_j]}(y), \]

where the \(c_j\)’s are constants to be determined. This is a piecewise constant approximation of \(p(y)\) and is called a histogram.

How do we determine the constants \(c_j\)? Each one of these constants is the probability that a sample of \(Y\) falls inside the bin, i.e.,

\[ c_j = p(b_{j-1}\le Y \le b_j). \]

Of course, we can write this probability as

\[ c_j = F(b_j) - F(b_{j-1}), \]

where \(F(y)\) is the CDF of \(Y\). Therefore, we can approximate the constants using our estimate of the CDF. In the notation of the previous section, we have that:

\[ \bar{c}_{j,N} := \bar{F}_N(b_j) - \bar{F}_N(b_{j-1}). \]

Of course, this is nothing more but:

\[ \bar{c}_{j,N} = \frac{\text{number of samples that fall in bin }[b_{j-1},b_j]}{N} \]

Putting everything together, our estimate for the PDF \(p(y)\) is:

\[ \hat{p}_{M,N}(y) = \sum_{j=1}^M\bar{c}_{j,N} 1_{[b_{j-1}, b_j]}(y), \]

which does converge to \(p(y)\) (in some sense) as both \(N\) and \(M\) go to infinity.

Example: 1D CDF

We will continue using the 1D test function of Example 3.4 [Robert and Casella, 2004]. Assume that \(X\sim\mathcal{U}([0,1])\) and pick:

\[ g(x) = \left(\cos(50x) + \sin(20x)\right)^2. \]

Show code cell source Hide code cell source

import numpy as np
# define the function here:
g = lambda x: (np.cos(50 * x) + np.sin(20 * x)) ** 2

# Again, we do not need to write any code for the histogram
# It's already implemented in several packages.
# We will use the matplotlib implementation

# Maximum number of samples to take
max_n = 10000 
# The number of bins
num_bins = 100
# Generate samples from X
x_samples = np.random.rand(max_n)
# Get the corresponding Y's
y_samples = g(x_samples)

# Make the plot
for N in [100, 1000, max_n]:
    fig, ax = plt.subplots()
    ax.hist(
        y_samples[:N],
        label=f"$N={N:d}$",
        bins=num_bins,
        density=True,
        alpha=0.25
    )
    ax.set_xlabel(r"$y$")
    ax.set_ylabel(r"$\hat{{p}}_{{M={0:d},N}}(y)$".format(num_bins))
    plt.legend(loc="best", frameon=False)
    sns.despine(trim=True);

Questions

• Experiment with the number of bins \(M\). Repeat the code above with \(M=5, 10\), and \(1000\). What do you observe? What happens when you have too few bins? What happens when you have too many bins? You should pick the number of bins and \(N\) together. As a rule, \(N\) should be about ten times \(M\). For a given choice of \(M\), it is possible to pick how many \(N\)’s you need using what we will learn in lecture 10.