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MAKE_BOOK_FIGURES=Trueimport numpy as npimport scipy.stats as stimport matplotlib as mplimport matplotlib.pyplot as plt%matplotlib inlineimport matplotlib_inlinematplotlib_inline.backend_inline.set_matplotlib_formats('svg')import seaborn as snssns.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()

Information Entropy

Information entropy is a measure of the uncertainty of a probability distribution. It is defined as:

\[ \mathbb{H}[p(X)] = -\sum_x \log p(x) p(x). \]

The sum is over all possible values of the random variable \(X\). If \(X\) is continuous, the sum becomes an integral:

\[ \mathbb{H}[p(X)] = -\int_x \log p(x) p(x) dx. \]

Example - Information Entropy of a Binary Distribution

Let’s take a random variable \(X\) with two possible values, say \(0\) and \(1\). Two numbers can describe the probability mass function:

\[ p_0 = p(X=0), \]

and

\[ p_1 = p(X=1) = 1 - p_0. \]

So, the information entropy of this distribution is simply a function of \(p_0\):

\[ \mathbb{H}[p(X)] = -\sum_x \log p(x) p(x) = -p_0 \log p_0 - p_1 \log p_1 = -p_0 \log p_0 + (1-p_0)\log (1-p_0). \]

Let’s plot it as we vary \(p_0\):

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eps = 1e-8
p = np.linspace(eps, 1. - eps, 100)
H = -p * np.log(p) - (1. - p) * np.log(1. - p)

fig, ax = plt.subplots()
ax.plot(p, H)
ax.set_xlabel('$p_0$')
ax.set_ylabel('Entropy')
sns.despine(trim=True);

Notice that the function is maximized at \(p_0 = 0.5\) because this corresponds to maximum uncertainty. The function is minimized (as a matter of fact it is exactly zero) at \(p_0 = 0\) and \(p_0 = 1\) because both these cases correspond to minimum uncertainty (you are certain what is going to happen).

Questions

• You are given two Categorical distributions:

\[ X\sim \operatorname{Categorical}(0.1, 0.3, 0.5, 0.1), \]

and

\[ Y\sim \operatorname{Categorical}(0.2, 0.2, 0.4, 0.2). \]

Let’s visualize them:

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import scipy.stats as st

X = st.rv_discrete(
    values=(
        np.arange(4),
        [0.1, 0.3, 0.5, 0.1]
    )
)
Y = st.rv_discrete(
    values=(
        np.arange(4),
        [0.2, 0.2, 0.4, 0.2]
    )
)

fig, ax = plt.subplots()
ax.bar(
    range(4),
    X.pmf(np.arange(4)),
    alpha=0.5,
    label='$X$'
)
ax.bar(
    range(4),
    Y.pmf(np.arange(4)),
    alpha=0.5,
    label='$Y$'
)
plt.legend(loc='best', frameon=False)
ax.set_xlabel('Values')
ax.set_ylabel('Probability')
sns.despine(trim=True);

Questions

• Based on the picture above which of the two random variables, \(X\) or \(Y\), has the most uncertainty?

• Use the block code below to calculate the entropy of each one of the distributions and answer the question above (which variable is more uncertaint) in a quantitative way. We can use the functionality of scipy.stats to compute the entropy.

ent_X = X.entropy()
print(f'H[X] = {ent_X:.2f}')
# Write code that computes and prints the entropy of Y

H[X] = 1.17