Explain Bayesian vs frequentist with an example

Question

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

Codemia · Accepted Answer

To understand the difference between Bayesian and frequentist approaches, we need to set up a scenario where we want to estimate the average time users spend on Dropbox per session. Assume we have data from a sample of 100 users where the average time spent is 30 minutes with a standard deviation of 10 minutes.

In a frequentist framework, we will calculate a 95% confidence interval for the mean time spent, while in a Bayesian framework, we will incorporate prior beliefs about the mean time spent (e.g., based on previous studies) and update this belief with our sample data.

1. **Frequentist Approach**: Calculate the 95% confidence interval using the formula:  
   $$ CI = \bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) $$  
   where $ \bar{x} $ is the sample mean, $ s $ is the standard deviation, $ n $ is the sample size, and $ z_{\alpha/2} $ is the z-value for 95% confidence (1.96).

2. **Bayesian Approach**: Assume a prior distribution for the mean (e.g., normal distribution with mean 25 and standard deviation 5). We will update this prior using Bayes' theorem to obtain the posterior distribution after observing our sample.

1. **Frequentist CI Calculation**:  
   - Sample mean ($ \bar{x} $): 30  
   - Standard deviation ($ s $): 10  
   - Sample size ($ n $): 100  
   - Standard error (SE):  $ SE = \frac{s}{\sqrt{n}} = \frac{10}{\sqrt{100}} = 1 $  
   - CI = 30 ± 1.96 * 1 = [28.04, 31.96].

2. **Bayesian Posterior Calculation**:  
   - Prior: Normal(25, 5)  
   - Likelihood: Normal(30, 10/sqrt(100))  
   - Posterior: The posterior will combine these distributions, resulting in a new mean and variance calculated using Bayesian updating formulas. Assume the posterior mean is approximately 29.5 with reduced variance.

The frequentist confidence interval [28.04, 31.96] means we are 95% confident that the true average time users spend on Dropbox per session falls within this range, based solely on the sample data. The Bayesian posterior mean of 29.5 suggests that after incorporating our prior belief, we believe the average is slightly lower than the sample mean but still close. The Bayesian method allows us to incorporate prior information, which can be beneficial if the prior is well-grounded.

This example connects to several related concepts:  
1. **Causal Inference**: Understanding how to differentiate correlation from causation can be explored further in both frameworks.  
2. **Central Limit Theorem (CLT)**: The frequentist method relies on the CLT for large samples, indicating that sample means will be normally distributed.  
3. **Power Analysis**: In the context of frequentist statistics, we might discuss how sample size affects the power of a test to detect differences.  
4. **Potential Confounders**: In both approaches, identifying and adjusting for confounders (e.g., user demographics) is crucial for valid inference.

Explain Bayesian vs frequentist with an example

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