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import numpy as np
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");

Credible Intervals

The posterior \(p(\theta|x_{1:N})\) captures everything we say about \(\theta\). Credible intervals are a way to summarize it. A credible interval is an interval inside which the parameter \(\theta\) lies with high probability. Specifically, a 95% credible interval \((\ell, u)\) (for lower and upper bounds) for \(\theta\) is such that:

\[ p(\ell \le \theta \le u|x_{1:N}) = 0.95. \]

Of course, there is not a unique, credible interval. You can move \((\ell, u)\) to the left or to the right in a way that keeps the probability contained in it at 0.95.

The central credible interval is particularly common. It is defined by solving the following root-finding problems:

\[ p(\theta \le \ell | x_{1:N}) = 0.025, \]

and

\[ p(\theta \le u | x_{1:N}) = 0.975, \]

for \(\ell\) and \(u\), respectively.

Let’s use a coin toss example to demonstrate this.

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import scipy.stats as st

theta_true = 0.8
X = st.bernoulli(theta_true)

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

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

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('$N={0:d}$'.format(N))
plt.legend(loc='best', frameon=False)
sns.despine(trim=True);

Here is how you can find the credible interval with the help of scipy.stats:

theta_low = Theta_post.ppf(0.025)
theta_up = Theta_post.ppf(0.975)
print(f'Theta is in [{theta_low:.2f}, {theta_up:1.2f}] with 95% probability')

Theta is in [0.36, 0.96] with 95% probability

Let’s visualize the credible interval:

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fig, ax = plt.subplots()
ax.plot(
    [theta_true],
    [0.0],
    'o',
    markeredgewidth=2,
    markersize=10,
    label='True value'
)
ax.plot(
    thetas,
    Theta_post.pdf(thetas),
    label=r'$p(\theta|x_{1:N})$'
)
thetas_int = np.linspace(theta_low, theta_up, 100)
ax.fill_between(
    thetas_int,
    np.zeros(thetas_int.shape),
    Theta_post.pdf(thetas_int),
    color='red',
    alpha=0.25
)
ax.plot(
    [theta_low, theta_up],
    np.zeros((2,)),
    'x',
    color='red',
    markeredgewidth=2,
    label='95% central CI'
)
ax.set_xlabel(r'$\theta$')
ax.set_title(f'$N={N}$')
plt.legend(loc='best', frameon=False)
sns.despine(trim=True);

So, is there another 95% credible interval? Yes there is. You can find it by solving thes problem:

\[ p(\theta \le \ell | x_{1:N}) = 0.01, \]

and

\[ p(\theta \le u | x_{1:N}) = 0.96, \]

for \(\ell\) and \(u\), respectively. Here is what you will find for the coin toss example.

theta_low_o = Theta_post.ppf(0.01)
theta_up_o = Theta_post.ppf(0.96)
print(f'Theta is in [{theta_low_o:.2f}, {theta_up_o:1.2f}] with 95% probability')

Theta is in [0.29, 0.94] with 95% probability

And here is how it compares to the previous one:

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fig, ax = plt.subplots()
ax.plot(
    [theta_true],
    [0.0],
    'o',
    markeredgewidth=2,
    markersize=10,
    label='True value'
)
ax.plot(
    thetas,
    Theta_post.pdf(thetas),
    label=r'$p(\theta|x_{1:N})$'
)
thetas_int = np.linspace(theta_low, theta_up, 100)
ax.fill_between(
    thetas_int,
    np.zeros(thetas_int.shape),
    Theta_post.pdf(thetas_int),
    color='red',
    alpha=0.25
)
ax.plot(
    [theta_low, theta_up],
    np.zeros((2,)),
    'x',
    color='red', 
    markeredgewidth=2,
    label='95% central CI'
)
thetas_int_o = np.linspace(theta_low_o, theta_up_o, 100)
ax.fill_between(
    thetas_int_o,
    np.zeros(thetas_int_o.shape),
    Theta_post.pdf(thetas_int_o),
    color='blue',
    alpha=0.25
)
ax.plot(
    [theta_low_o, theta_up_o],
    np.zeros((2,)),
    '+',
    color='blue',
    markeredgewidth=2,
    label='Other 95% CI'
)
ax.set_xlabel(r'$\theta$')
ax.set_title(f'$N={N:d}$')
plt.legend(loc='best', frameon=False)
sns.despine(trim=True);

Getting Credible Intervals when the Posterior is not Analytically Available

Of course, you often do not have the posterior in analytical form and have to estimate the credible intervals via sampling. We will learned about this in Lecture 10.

Questions

• Find the credible interval for \(\theta\) conditioned on the data with 99% accuracy.

• How many coin tosses do you have to do to estimate \(\theta\) within an accuracy of \(1\%\) with \(99\%\) probability? Do not try to do this analytically. Just experiment with different values of \(N\) for this synthetic example. Getting a number \(N\) that works for all possible datasets (assuming the model is correct) is an exciting but not trivial problem.