#### Using Machine Learning for Dynamic Micro-Segmentation

Traditional customer segmentation methods are no longer sufficient to capture the nuances of customer behavior and preferences. Enter dynamic...

4 min read

Writing Team : Oct 15, 2024 11:41:08 AM

A/B testing has become an indispensable tool for data-driven decision-making. However, the traditional frequentist approach to A/B testing has limitations that can lead to suboptimal decisions, especially in the fast-paced world of growth marketing. Enter Bayesian statistics – a powerful alternative that offers more nuanced, flexible, and actionable insights. This article delves deep into the implementation of Bayesian statistics in A/B testing, providing expert marketers with the knowledge to make more accurate growth decisions.

Before we dive into Bayesian methods, let's briefly recap the limitations of traditional frequentist A/B testing:

**Fixed sample sizes**: Frequentist tests often require predefined sample sizes, which can be inefficient in dynamic marketing environments.**Binary outcomes**: Traditional tests typically provide a "significant" or "not significant" result, lacking nuance.**Misinterpretation of p-values**: P-values are often misunderstood, leading to poor decision-making.**Inability to incorporate prior knowledge**: Frequentist methods don't allow for the integration of historical data or expert intuition.

Bayesian A/B testing addresses these limitations by:

**Allowing for flexible sample sizes**: Tests can be stopped or continued as needed without compromising statistical validity.**Providing probability distributions**: Instead of binary outcomes, Bayesian methods offer probability distributions of possible effects.**Incorporating prior knowledge**: Historical data and expert intuition can be formally integrated into the analysis.**Offering more intuitive interpretations**: Results are expressed as probabilities of an effect, which are easier to understand and act upon.

Let's walk through it.

First, clearly define your key performance indicators (KPIs) and formulate your hypotheses. For example:

- KPI: Conversion Rate
- Null Hypothesis (H0): The new design (B) has no effect on the conversion rate compared to the current design (A).
- Alternative Hypothesis (H1): The new design (B) increases the conversion rate compared to the current design (A).

One of the key advantages of Bayesian methods is the ability to incorporate prior knowledge. This is done through specifying prior distributions for your parameters of interest. For conversion rates, a Beta distribution is often used due to its properties and conjugacy with the Binomial distribution.

Example: Let's say your historical data shows that your conversion rate typically ranges between 2% and 5%. You might specify a Beta(10, 290) as your prior, which has a mean of 3.33% and a 95% credible interval of [1.6%, 5.7%].

`from scipy.stats import beta`

# Define prior

alpha_prior = 10

beta_prior = 290

# Plot prior distribution

x = np.linspace(0, 0.1, 1000)

plt.plot(x, beta.pdf(x, alpha_prior, beta_prior))

plt.title("Prior Distribution for Conversion Rate")

plt.xlabel("Conversion Rate")

plt.ylabel("Density")

plt.show()

Run your A/B test and collect data. Let's say after running the test for a week, you have:

- Control (A): 100 conversions out of 3000 visitors
- Variant (B): 120 conversions out of 3000 visitors

Now, use Bayes' theorem to update your prior beliefs with the observed data. The posterior distribution for each variant will be:

- Posterior_A = Beta(α_prior + conversions_A, β_prior + visitors_A - conversions_A)
- Posterior_B = Beta(α_prior + conversions_B, β_prior + visitors_B - conversions_B)

# Update posteriors

`alpha_posterior_A = alpha_prior + 100`

beta_posterior_A = beta_prior + 3000 - 100

alpha_posterior_B = alpha_prior + 120

beta_posterior_B = beta_prior + 3000 - 120

# Plot posterior distributions

plt.plot(x, beta.pdf(x, alpha_posterior_A, beta_posterior_A), label='A')

plt.plot(x, beta.pdf(x, alpha_posterior_B, beta_posterior_B), label='B')

plt.title("Posterior Distributions for Conversion Rates")

plt.xlabel("Conversion Rate")

plt.ylabel("Density")

plt.legend()

plt.show()

With Bayesian methods, we can answer questions like:

- What's the probability that B is better than A?
- What's the expected lift of B over A?
- What's the 95% credible interval for the difference between B and A?

# Probability that B is better than A

`samples_A = beta.rvs(alpha_posterior_A, beta_posterior_A, size=100000)`

samples_B = beta.rvs(alpha_posterior_B, beta_posterior_B, size=100000)

prob_B_better = np.mean(samples_B > samples_A)

print(f"Probability that B is better than A: {prob_B_better:.2%}")

# Expected lift

expected_lift = np.mean(samples_B) / np.mean(samples_A) - 1

print(f"Expected lift of B over A: {expected_lift:.2%}")

# 95% credible interval for the difference

diff_samples = samples_B - samples_A

credible_interval = np.percentile(diff_samples, [2.5, 97.5])

print(f"95% credible interval for difference: [{credible_interval[0]:.4f}, {credible_interval[1]:.4f}]")

Based on these results, you can make more informed decisions. For example:

- If the probability that B is better than A is 95% or higher, you might decide to implement B.
- If the expected lift is substantial (e.g., >5%) but the probability is only 80%, you might decide to continue the test to gather more data.
- If the 95% credible interval includes 0 but is skewed positive, you might decide to implement B if the cost of implementation is low.

Let's go a little deeper.

Bayesian methods naturally extend to multi-armed bandit algorithms, which can dynamically allocate traffic to better-performing variants during the test. This can be particularly valuable for short-lived campaigns or when opportunity costs are high.

Example implementation using Thompson Sampling:

import numpy as np

`from scipy.stats import beta`

def thompson_sampling(alpha_A, beta_A, alpha_B, beta_B):

sample_A = beta.rvs(alpha_A, beta_A)

sample_B = beta.rvs(alpha_B, beta_B)

return 'A' if sample_A > sample_B else 'B'

# Simulate 10000 visitors

for _ in range(10000):

chosen_variant = thompson_sampling(alpha_posterior_A, beta_posterior_A,

alpha_posterior_B, beta_posterior_B)

# Simulate conversion (you'd replace this with actual data in a real scenario)

converted = np.random.random() < (0.033 if chosen_variant == 'A' else 0.04)

# Update posteriors

if chosen_variant == 'A':

alpha_posterior_A += converted

beta_posterior_A += 1 - converted

else:

alpha_posterior_B += converted

beta_posterior_B += 1 - converted

# Final results

print(f"A: Beta({alpha_posterior_A}, {beta_posterior_A})")

print(f"B: Beta({alpha_posterior_B}, {beta_posterior_B})")

For businesses running multiple related tests (e.g., across different geographic regions), hierarchical Bayesian models can pool information across tests, leading to more accurate estimates, especially for segments with limited data.

Example using PyMC3:

import pymc3 as pm

# Assume we have data from 5 regions

conversions_A = [95, 80, 100, 90, 110]

visitors_A = [3000, 2500, 3100, 2800, 3200]

conversions_B = [110, 95, 120, 105, 130]

visitors_B = [3000, 2500, 3100, 2800, 3200]

with pm.Model() as hierarchical_model:

# Hyperpriors

mu_alpha = pm.Normal('mu_alpha', mu=0, sd=1)

sigma_alpha = pm.HalfNormal('sigma_alpha', sd=1)

# Region-specific effects

alpha = pm.Normal('alpha', mu=mu_alpha, sd=sigma_alpha, shape=5)

# Treatment effect

beta = pm.Normal('beta', mu=0, sd=1)

# Conversion rates

theta_A = pm.Deterministic('theta_A', pm.math.invlogit(alpha))

theta_B = pm.Deterministic('theta_B', pm.math.invlogit(alpha + beta))

# Likelihood

y_A = pm.Binomial('y_A', n=visitors_A, p=theta_A, observed=conversions_A)

y_B = pm.Binomial('y_B', n=visitors_B, p=theta_B, observed=conversions_B)

# Inference

trace = pm.sample(2000, tune=1000)

# Analyze results

pm.plot_posterior(trace, var_names=['beta'])

Implementing Bayesian statistics in A/B testing offers a more nuanced, flexible, and powerful approach to making growth decisions. By incorporating prior knowledge, providing probabilistic outcomes, and allowing for more intuitive interpretation of results, Bayesian methods enable marketers to make better-informed decisions in dynamic environments.

As you implement these methods, remember:

- Clearly define your metrics and hypotheses.
- Thoughtfully specify your priors based on historical data and expert knowledge.
- Continuously update your beliefs as new data comes in.
- Make decisions based on probabilities and expected values, not just point estimates.
- Consider advanced techniques like multi-armed bandits and hierarchical models for more complex scenarios.

By mastering Bayesian A/B testing, you'll be equipped to navigate the complexities of modern growth marketing with greater precision and confidence.

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