Define x-axis (conversion rate from 0 to 1)

When implementing Bayesian methods, the most critical step is to clearly separate prior assumptions from empirical evidence—and then assess whether the resulting posterior provides sufficient support for decision-making.

I ask: Where does the prior come from? Is the sample biased? What is the cost of making an incorrect decision?

In the previous chapter, we discussed Bayesian updating and the roles of prior and posterior distributions—laying a statistical foundation for the data analysis that follows. In this chapter, we demonstrate how to apply those theoretical concepts through a concrete, real-world case study—performing effective analysis and interpreting results meaningfully.

Case Background

Suppose we are an online retail company deeply interested in customer purchasing behavior. To improve conversion rates, we launch a market research initiative evaluating how different advertising strategies influence customers’ purchase decisions—using a Bayesian approach.

When working through a Bayesian data analysis case, first examine: the problem hypothesis, the source of the prior, the observed data, the likelihood model, the posterior result, and sensitivity checks.

Data Collection

We collect data via an online survey, assigning respondents to two groups:

• Group A: Exposed to traditional advertising (e.g., TV, newspapers)
• Group B: Exposed to digital advertising (e.g., social media, search engines)

Each group contains 100 participants; for each individual, we record whether they made a purchase (“Yes” or “No”).

Summary of Results

Group	Purchased	Did Not Purchase	Total
A	30	70	100
B	50	50	100

Using this data, we apply Bayesian updating to assess the effectiveness of the experiment.

Bayesian Updating

We adopt a Beta distribution as our prior: $\text{Beta}(1, 1)$, representing a uniform (non-informative) belief—i.e., no preference before seeing any data.

While reading Practical Data Analysis Cases: From Bayesian Theory to Applied Practice, treat the accompanying figures as roadmap cards: First grasp the overall workflow; then understand why each step is performed; finally verify boundary conditions and assumptions.

Computing the Posterior Distributions

The posterior for each group updates the prior using the observed counts:

1. For Group A:

   • Purchased = 30, Did Not Purchase = 70
   • Posterior = $\text{Beta}(30 + 1,\, 70 + 1) = \text{Beta}(31,\, 71)$

2. For Group B:

   • Purchased = 50, Did Not Purchase = 50
   • Posterior = $\text{Beta}(50 + 1,\, 50 + 1) = \text{Beta}(51,\, 51)$

Visualizing the Posterior Distributions

We visualize these posteriors using Python and Matplotlib:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import beta

# Define x-axis (conversion rate from 0 to 1)
x = np.linspace(0, 1, 100)
a_A, b_A = 31, 71
a_B, b_B = 51, 51

# Compute probability density functions
y_A = beta.pdf(x, a_A, b_A)
y_B = beta.pdf(x, a_B, b_B)

# Plot
plt.plot(x, y_A, label='Group A Posterior (Beta(31, 71))')
plt.plot(x, y_B, label='Group B Posterior (Beta(51, 51))')
plt.title('Posterior Distribution Comparison')
plt.xlabel('Purchase Conversion Rate')
plt.ylabel('Probability Density')
plt.legend()
plt.grid()
plt.show()

Interpreting the Results

From the posterior distributions, we compare our updated “beliefs” about purchase conversion rates for the two groups. The mode (peak) and spread of each distribution reveal that Group B’s advertising strategy is more likely to drive purchases—reflected in both a higher central tendency and greater concentration around higher conversion rates.

After completing Practical Data Analysis Cases: From Bayesian Theory to Applied Practice, try adapting it to your own scenario. Focus especially on whether inputs, processing steps, and outputs align coherently.

To apply Practical Data Analysis Cases: From Bayesian Theory to Applied Practice to your own task, start small: isolate and validate just one critical decision point.

Conclusion

Bayesian updating not only yields intuitive insights into advertising effectiveness—it also equips decision-makers with quantified, probabilistic support grounded in the posterior distribution. Next, we will explore how to evaluate and select among models to rigorously validate both the analytical method and its conclusions.

In upcoming chapters, we’ll focus specifically on measuring model accuracy and precision—enabling us to refine and optimize our advertising strategy.