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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()

Example: Inferring the probability of a coin toss from data

We toss a coin with an unknown probability of heads \(\theta\) \(N\) times independently, and we observe the result:

\[ x_{1:N} := (x_1,\dots,x_N). \]

Assume that we have coded the result so that heads correspond to a “1” and tails to a “0.” We aim to estimate the probability of heads \(\theta\) from this dataset.

Assuming that we know nothing, we set:

\[ \theta \sim U([0,1]). \]

In terms of probability densities, this is:

\[\begin{split} p(\theta) = \begin{cases} 1,&\;\text{if}\;\theta\in [0,1].\\ 0,&\;\text{otherwise} \end{cases} \end{split}\]

Now, let’s write down the likelihood of the data. Because of the independence assumption, we have:

\[ p(x_{1:N}|\theta) = \prod_{n=1}^Np(x_n|\theta). \]

Then, each measurement is a Bernoulli with probability of success \(\theta\), i.e.,

\[ x_n | \theta \sim \operatorname{Bernoulli}(\theta). \]

In terms of probability densities, we have the likelihood:

\[\begin{split} p(x_n|\theta) = \begin{cases} \theta,&\;\text{if}\;x_n=1,\\ 1-\theta,&\;\text{otherwise}. \end{cases} \end{split}\]

Using a common mathematical trick, we can rewrite this as:

\[ p(x_n|\theta) = \theta^{x_n}(1-\theta)^{1-x_n}. \]

Work out the cases \(x_n=0\) and \(x_n=1\) to convince yourself.

Now, we can find the expression for the likelihood of the entire dataset. It is:

\[\begin{split} \begin{split} p(x_{1:N}|\theta) &= \prod_{n=1}^Np(x_n|\theta)\\ &= \prod_{n=1}^N\theta^{x_n}(1-\theta)^{1-x_n}\\ &= \theta^{\sum_{n=1}^Nx_n}(1-\theta)^{N-\sum_{n=1}^Nx_n}. \end{split} \end{split}\]

This intuitively means the probability of getting \(\sum_{n=1}^Nx_n\) heads and the rest \(N-\sum_{n=1}^Nx_n\) tails.

We can now find the posterior. It is:

\[ \text{posterior} \propto \text{likelihood}\times\text{prior}. \]

In our problem:

\[\begin{split} \begin{split} p(\theta|x_{1:N}) &\propto p(x_{1:N}|\theta)p(\theta)\\ &= \begin{cases} \theta^{\sum_{n=1}^Nx_n}(1-\theta)^{N-\sum_{n=1}^N},&\;\text{if}\;\theta\in[0,1]\\ 0,&\;\text{otherwise}. \end{cases} \end{split} \end{split}\]

And this is just the density corresponding to a Beta distribution:

\[ p(\theta|x_{1:N}) = \operatorname{Beta}\left(\theta\middle|1 + \sum_{n=1}^Nx_n, 1 + N - \sum_{n=1}^Nx_n\right). \]

Let’s try this out with some fake data. Take a fake coin which is a little bit biased:

import scipy.stats as st
theta_true = 0.8
X = st.bernoulli(theta_true)

Sample from it a number of times to generate our data = (x1, …, xN):

N = 5
data = X.rvs(size=N)
data

array([0, 1, 1, 1, 1])

Now we are ready to calculate the posterior which the Beta we have above:

alpha = 1.0 + data.sum()
beta = 1.0 + N - data.sum()
Theta_post = st.beta(alpha, beta)

And we can plot it:

fig, ax = plt.subplots()
thetas = np.linspace(0, 1, 100)
ax.plot(
    [theta_true],
    [0.0],
    'o',
    markeredgewidth=2,
    markersize=10,
    label='True value'
)
ax.plot(
    thetas,
    Theta_post.pdf(thetas),
    label=r'$p(\theta|x_{1:N})$'
)
ax.set_xlabel(r'$\theta$')
ax.set_ylabel('Probability density')
ax.set_title(f'$N={N}$')
plt.legend(loc='best', frameon=False)
sns.despine(trim=True);

Questions

• Try \(N=0,5,10,100\) and see what happens.