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MAKE_BOOK_FIGURES=True
import numpy as np
import scipy.stats as st

import matplotlib as mpl
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")

def set_book_style():
    plt.style.use('seaborn-v0_8-white') 
    sns.set_style("ticks")
    sns.set_palette("deep")

    mpl.rcParams.update({
        # Font settings
        'font.family': 'serif',  # For academic publishing
        'font.size': 8,  # As requested, 10pt font
        'axes.labelsize': 8,
        'axes.titlesize': 8,
        'xtick.labelsize': 7,  # Slightly smaller for better readability
        'ytick.labelsize': 7,
        'legend.fontsize': 7,
        
        # Line and marker settings for consistency
        'axes.linewidth': 0.5,
        'grid.linewidth': 0.5,
        'lines.linewidth': 1.0,
        'lines.markersize': 4,
        
        # Layout to prevent clipped labels
        'figure.constrained_layout.use': True,
        
        # Default DPI (will override when saving)
        'figure.dpi': 600,
        'savefig.dpi': 600,
        
        # Despine - remove top and right spines
        'axes.spines.top': False,
        'axes.spines.right': False,
        
        # Remove legend frame
        'legend.frameon': False,
        
        # Additional trim settings
        'figure.autolayout': True,  # Alternative to constrained_layout
        'savefig.bbox': 'tight',    # Trim when saving
        'savefig.pad_inches': 0.1   # Small padding to ensure nothing gets cut off
    })

def set_notebook_style():
    plt.style.use('seaborn-v0_8-white')
    sns.set_style("ticks")
    sns.set_palette("deep")

    mpl.rcParams.update({
        # Font settings - using default sizes
        'font.family': 'serif',
        'axes.labelsize': 10,
        'axes.titlesize': 10,
        'xtick.labelsize': 9,
        'ytick.labelsize': 9,
        'legend.fontsize': 9,
        
        # Line and marker settings
        'axes.linewidth': 0.5,
        'grid.linewidth': 0.5,
        'lines.linewidth': 1.0,
        'lines.markersize': 4,
        
        # Layout settings
        'figure.constrained_layout.use': True,
        
        # Remove only top and right spines
        'axes.spines.top': False,
        'axes.spines.right': False,
        
        # Remove legend frame
        'legend.frameon': False,
        
        # Additional settings
        'figure.autolayout': True,
        'savefig.bbox': 'tight',
        'savefig.pad_inches': 0.1
    })

def save_for_book(fig, filename, is_vector=True, **kwargs):
    """
    Save a figure with book-optimized settings.
    
    Parameters:
    -----------
    fig : matplotlib figure
        The figure to save
    filename : str
        Filename without extension
    is_vector : bool
        If True, saves as vector at 1000 dpi. If False, saves as raster at 600 dpi.
    **kwargs : dict
        Additional kwargs to pass to savefig
    """    
    # Set appropriate DPI and format based on figure type
    if is_vector:
        dpi = 1000
        ext = '.pdf'
    else:
        dpi = 600
        ext = '.tif'
    
    # Save the figure with book settings
    fig.savefig(f"{filename}{ext}", dpi=dpi, **kwargs)

def make_full_width_fig():
    return plt.subplots(figsize=(4.7, 2.9), constrained_layout=True)

def make_half_width_fig():
    return plt.subplots(figsize=(2.35, 1.45), constrained_layout=True)

if MAKE_BOOK_FIGURES:
    set_book_style()
else:
    set_notebook_style()

make_full_width_fig = make_full_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()
make_half_width_fig = make_half_width_fig if MAKE_BOOK_FIGURES else lambda: plt.subplots()

Sampling from continuous distributions - Inverse sampling

How do you sample an arbitrary univariate continuous random variable \(X\) with CDF \(F(x)\)? In this scenario, inverse sampling is the way to go. It relies on the observation that the random variable

\[ Y = F^{-1}(U), \]

where \(F^{-1}\) is the inverse of the CDF of \(X\) and \(U\sim\mathcal{U}([0,1])\) has exactly the same distribution as \(X\).

Proof

It suffice to show that \(Y\) has the same CDF as \(X\). Here we go:

\[\begin{split} \begin{align*} p(Y\le y) &= p(F^{-1}(U)\le x)\\ &= p(U\le F(x))\\ &= F(x), \end{align*} \end{split}\]

where in the last step we used the fact that \(U\sim\mathcal{U}([0,1])\) and therefore \(p(U\le u) = u\).

We will demonstrate this by example. To this end, let us consider an exponential random variable:

\[ X \sim \operatorname{Exp}(r), \]

where \(r > 0\) is known as the rate parameter. The exponential distribution describes the time between random events that occur at a constant rate \(r\). The CDF of the Exponential is:

\[ F(x) = p(X\le x) = 1 - e^{-rx}. \]

Let’s plot it for \(r=0.5\).

import scipy.stats as st
import numpy as np

r = 0.5
X = st.expon(scale=1.0 / r)

fig, ax = plt.subplots()
x = np.linspace(0., 5. / r, 100)
ax.plot(x, X.cdf(x))
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$F(x) = p(X \leq x)$")
ax.set_title(f"$X\sim E(r={r:1.2f})$")
sns.despine(trim=True);

To sample \(X\) using inverse sampling, we need the inverse of the CDF. This is easily shown to be:

\[ F^{-1}(u) = -\frac{\ln(1-u)}{r}. \]

Let’s see if this is going to give us the right samples. We will compare the empirical histogram obtained by inverse sampling to the actual PDF \(p(x)\). Here is the code for inverse sampling:

def sample_exp(r : float):
    """Sample from an exponential.
    
    Arguments:
    r  --  The rate parameter.
    """
    u = np.random.rand()
    return -np.log(1. - u) / r

And here is the histogram of some samples:

N = 10000
x_samples = np.array(
    [sample_exp(r) for _ in range(N)]
)

fig, ax = plt.subplots()
ax.hist(
    x_samples,
    alpha=0.5,
    density=True,
    bins=100,
    label="Histogram of samples"
)
ax.plot(x, X.pdf(x))
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$p(x)$")
plt.legend(loc=r"best", frameon=False)
sns.despine(trim=True);

Questions

• Modify the code above to implement inverse sampling for a univariate Gaussian with zero mean and unit variance. Use scipy.stats to find the inverse CDF of the Gaussian (It is st.norm.ppf). Here is how to use it:

# Standard normal random variable
Z = st.norm(loc=0.0, scale=1.0)
# The inverse CDF of the standard normal, say at 0.7, can be evaluated by:
Z.ppf(0.7)

0.5244005127080407