Bayesian Coin

Imports¶

Part 1: Probability Distributions¶

Cell 1.1: Bernoulli¶

Goal:

• Build intuition for the Bernoulli distribution, a single trial with two outcomes
• $X \sim \text{Bernoulli}(p)$ means $P(X = 1) = p$ and $P(X = 0) = 1 - p$, $0 \leq p \leq 1$

Implementation: cell1_1_sample_bernoulli_widget()

• Draws n samples from scipy.stats.bernoulli(p) and prints them

Usage

• Inputs

  • n: number of samples drawn, 1-50
  • p: success probability, 0-1

• Panels

  • data: printed array of n realizations, each 0 or 1

Guided usage

• Raise p from 0 toward 1, leaving n fixed
  • Observe the printed data shift from mostly 0s to mostly 1s: a coin flip is $X=1$ if heads, $X=0$ if tails, and p is the heads probability

Cell 1.2: Binomial¶

Goal:

• Build intuition for the binomial distribution, the count of successes over trials independent Bernoulli draws
• $P(X = k) = \binom{\text{trials}}{k} p^k (1 - p)^{\text{trials} - k}$

Implementation: cell1_2_sample_binomial_widget()

• Draws n samples from scipy.stats.binom(trials, p) and prints them

Usage

• Inputs

  • n: number of samples drawn, 1-50
  • trials: trials per sample, 1-100
  • p: success probability, 0-1

• Panels

  • data: printed array of n realizations, each a count in [0, trials]

Guided usage

• Raise trials from 1 toward 100, leaving p fixed
  • Observe the printed counts spread over a wider range: e.g. flipping a fair coin 10 times, the number of heads follows Binomial(10, 0.5)

Cell 1.3: Interactive binomial PMF¶

Goal:

• Explore the binomial PMF’s shape directly through preliz’s own interactive widget

Implementation: pz.Binomial(p, n).plot_interactive(**params)

• preliz builds and displays its own sliders for p and n; kind and interval control how the distribution is summarized

Cell 1.4: Binomial distribution grid¶

Goal:

• Compare the binomial PMF’s shape across a grid of n and p values at once

Implementation: plot_binomial()

• Plots stats.binom(n, p).pmf(x) for every combination of 3 n values and 5 p values, one subplot each

Cell 1.5: Beta¶

Goal:

• Build intuition for the Beta distribution, a continuous distribution on $[0, 1]$ used to model a probability or proportion
• a is the “success” shape parameter, b is the “failure” shape parameter: a > b skews the density toward 1, a = b centers it on 0.5

Implementation: cell1_5_sample_beta_widget()

• Draws n samples from scipy.stats.beta(a, b) and prints them

Usage

• Inputs

  • n: number of samples drawn, 1-50
  • a: alpha shape parameter, 0.1-10
  • b: beta shape parameter, 0.1-10

• Panels

  • data: printed array of n realizations in [0, 1]

Guided usage

• Set a well above b, then well below it
  • Observe the printed samples cluster near 1 in the first case and near 0 in the second: a > b skews toward 1, a < b skews toward 0

Cell 1.6: Interactive beta PMF¶

Goal:

• Explore the Beta density’s shape directly through preliz’s own interactive widget

Implementation: pz.Beta(alpha, beta).plot_interactive(**params)

• preliz builds and displays its own sliders for alpha and beta

Cell 1.7: Beta distribution grid¶

Goal:

• Compare the Beta density’s shape across a grid of alpha and beta values at once

Implementation: plot_beta()

• Plots stats.beta(a, b).pdf(x) for every combination of 4 alpha values and 4 beta values, one subplot each

Part 2: Coin Example, Analytical Solution¶

Cell 2.1: Beta prior updated by streaming data¶

Goal:

• Watch a single Beta prior turn into a posterior as more coin-flip data arrives, one trial count at a time
• prior=(1, 1) is uniform, (20, 20) looks Gaussian centered on 0.5, (1, 4) looks exponential centered near 0

Implementation: beta_prior_interactive()

• Generates binomial counts $y \sim \text{Binomial}(N, \theta)$ at each n_trials entry with _generate_data()
• Plots the Beta(a+y, b+N-y) posterior density for the trial count at Index, steppable via the play button

Usage

• Inputs

  • seed: random seed for the generated data
  • theta (true): the true, otherwise-unknown success probability
  • n_trials: comma-separated trial counts to step through
  • a, b: the prior’s Beta shape parameters
  • Index/play button: which n_trials entry is shown

• Panels

  • Posterior after N=... trials, y=... heads: the posterior density, with theta (true) marked
  • Comments: current parameters and the resulting posterior distribution

Guided usage

• Set a/b to 1, 1 (uniform prior), then step Index forward
  • Observe the posterior widen out then narrow around theta (true) as N grows
• Compare a/b = 1, 1 against 20, 20 at the same Index
  • Observe the strong (20, 20) prior barely moves at low N, while the uniform prior is dominated by the data almost immediately

Cell 2.2: Updating three priors at once¶

Goal:

• Compare three different Beta priors, uniform, Gaussian-like, and exponential-like, updated by the same growing dataset side by side

Implementation: update_prior()

• Fixes theta_real = 0.35 and a shared sequence of n_trials/observed head counts
• Plots all three priors’ posteriors at each trial count in one grid, so every prior can be compared at the same amount of data

• All three posteriors converge toward the same spike at theta_real as N grows, regardless of how different the priors started
• The Bayesian update replaces “guess and check” with an explicit rule for how a prior belief and new data combine into a posterior belief

Part 3: Coin Example, Numerical Solution¶

• It’s a synthetic example!

  • Assume you know the true value of $\theta$ (not true in general)

• Workflow

  • Model the prior $\theta$ and the likelihood $Y \mid \theta$
    $$\begin{cases} \theta \sim \text{Beta}(\alpha = 1, \beta = 1) \\ Y \sim \text{Binomial}(n = 1, p = \theta) \\ \end{cases}$$
  • Observe samples of the variable $Y$
  • Run inference
  • Generate samples of the posterior
  • Summarize the posterior
    • E.g., Highest-Posterior Density (HPD)

Cell 3.1: Fitting the model with N=4¶

Goal:

• Fit the model above to N=4 synthetic observations, and read the fitted posterior with arviz’s trace, summary, rank-bar, and posterior plots

Implementation:

• Builds model1 with a Beta(1, 1) prior and a Bernoulli likelihood observing data1, then draws posterior samples with pm.sample() (PyMC’s NUTS sampler, 4 chains)
• az.plot_trace()/az.summary()/az.plot_posterior() summarize the resulting idata1

• No trace diverges, and the 4 chains mix well
• The KDE for theta should resemble a Beta density

• $E[\hat{\theta}] \approx 0.324$
• $\Pr(\hat{\theta} \in [0.031, 0.653]) = 0.94$

Cell 3.2: More data (N=20)¶

Goal:

• Repeat Cell 3.1 with N=20 observations, and see the posterior sharpen around theta_real as more data arrives

Implementation:

• Same Beta(1, 1)/Bernoulli model as model1, refit on 20 observations as model2

Cell 3.3: Even more data (N=100)¶

Goal:

• Repeat Cell 3.1 with N=100 observations, to see the posterior sharpen even further

Implementation:

• Same Beta(1, 1)/Bernoulli model as model1, refit on 100 observations as model3

Cell 3.4: Savage-Dickey ratio¶

Goal:

• Test the point hypothesis $\theta = 0.5$ against each of the three fitted posteriors via the Savage-Dickey density ratio

Implementation:

• az.plot_bf() compares the posterior density at ref_val=0.5 against a uniform prior, for idata1, idata2, and idata3 in turn

Cell 3.5: ROPE¶

Goal:

• Test whether $\theta$ plausibly equals 0.5 using a Region Of Practical Equivalence (ROPE) instead of a single point

Implementation:

• az.plot_posterior() shades the [0.45, 0.55] ROPE and reports the fraction of the posterior it contains, for each fitted idata

Part 4: Decision With a Loss Function¶

Cell 4.1: Minimizing a loss function¶

Goal:

• Turn a posterior into a point estimate by minimizing expected loss, instead of just reporting the posterior mean

Implementation: plot_loss(grid, loss_func)

• Plots the chosen loss_func (squared, absolute, asymmetric, or sine) against a candidate-estimate grid

Cell 4.2: Posterior samples as a distribution¶

Goal:

• Look at the raw posterior draws for theta directly: as a sequence, as a trace, and as a density

Implementation:

• Extracts the theta column from idata1.to_dataframe() and plots it as a trace and a KDE

Cell 4.3: Picking the best theta¶

Goal:

• Combine the loss function from Cell 4.1 with the posterior samples to pick the loss-minimizing point estimate, for both the N=4 and N=20 posteriors

Implementation: pick_best_theta(idata)

• Searches over candidate theta values and reports the one minimizing the expected loss under the posterior in idata