Explain Bayesian vs frequentist with an example

Question

Explain Bayesian vs frequentist in simple terms and provide a concrete example.

Codemia · Accepted Answer

To explore the differences between Bayesian and frequentist approaches, we can frame our analytical question as follows: "Does a new feature improve user engagement in our AI platform?"

For this analysis, we would need data on user engagement metrics before and after the feature's implementation. Specifically, we require:
- Pre-implementation engagement data (e.g., average session duration, number of interactions per session).
- Post-implementation engagement data.
- User demographic information to control for varying user behaviors.

The success metric could be defined as the percentage increase in average session duration after the feature rollout.

To analyze the data, we can use a Bayesian approach, which allows us to incorporate prior beliefs about user engagement and update those beliefs with observed data. We will set a prior distribution for the average user engagement based on historical data (e.g., a normal distribution with a mean of 5 minutes and a standard deviation of 1 minute).

Using a Bayesian model, we will compute the posterior distribution of the average engagement after the feature is introduced. This involves using the observed post-implementation data to update our prior beliefs, which can be done using tools like PyMC3 or Stan.

Imagine our prior belief about user engagement is modeled as: 
- Prior: N(5, 1)

After implementing the feature, we collect post-implementation data from 100 users and find that their average engagement is 6 minutes with a standard deviation of 1.5 minutes. We can then use Bayesian updating to combine our prior with the observed data.

Assuming a normal likelihood, our posterior distribution can be calculated using the following formula:

Posterior = (Prior * Likelihood) / Evidence.

Using a Bayesian updating formula, we might find that our posterior distribution is N(5.8, 0.3), indicating a new mean engagement of 5.8 minutes.

The posterior mean of 5.8 minutes suggests that, after implementing the new feature, user engagement has improved. We can also calculate the credible interval (e.g., 95%) for our posterior distribution, which may range from 5.4 to 6.2 minutes. This means we are 95% confident that the true average user engagement lies within this range.

In contrast, a frequentist approach would involve hypothesis testing, where we might set up a null hypothesis stating that the feature has no effect on engagement and conduct a t-test to determine if the observed changes are statistically significant.

Common mistakes in this type of analysis include:
1. **Ignoring Prior Distributions**: Not selecting appropriate priors based on historical data can lead to biased results.
2. **Misinterpreting Credible Intervals**: Confusing credible intervals with confidence intervals, which can lead to incorrect conclusions about the strength of evidence.
3. **Sample Size Concerns**: Not considering the sample size when interpreting results can lead to overconfidence in findings, especially if the sample is too small to draw robust conclusions.
4. **Neglecting Guardrail Metrics**: Failing to define and monitor guardrail metrics can result in overlooking potential negative consequences of feature changes.

Explain Bayesian vs frequentist with an example

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