Let’s talk about two of the common approaches to statistics today.
- Probability:
- Frequentist Statistics (FS), probability is interpreted as nothing but the long-run frequency of events in a repeated, hypothetical infinite sequence of trials. It’s based on the idea of objective randomness.
- Bayesian statistics (BS) views probability as a measure of belief or uncertainty. It incorporates prior beliefs and updates them based on new evidence using Bayes’ theorem.
- Parameter estimation:
- FS: The focus is on estimating fixed, unknown parameters from observed data. This estimation is done using methods like maximum likelihood estimation (MLE).
- BS: Bayesian inference provides a probability distribution for the parameters, incorporating prior knowledge and updating it with observed data to get a posterior distribution.
- Hypothesis testing:
- FS: Frequentist hypothesis testing involves making decisions about population parameters based on sample data. It often uses p-values to determine the level of significance.
- BS: Bayesian hypothesis testing involves comparing the probabilities of different hypotheses given the data. It uses posterior probabilities and Bayes factors to make decisions.
I used a Bayesian t-test strategy to take this prior knowledge into account because I firmly believe that the difference in average ages is approximately 7 and that it is statistically significant. The results, however, revealed an intriguing discrepancy: the observed difference was located towards the posterior distribution’s tail. I did not appreciate this disparity and how it demonstrated how sensitive Bayesian analysis is to previous specifications
To investigate the difference in average ages between black and white guys further, I used frequentist statistical methods such as Welch’s t-test in addition to the Bayesian approaches. The Welch’s t-test findings showed a very important finding:
In addition to the Bayesian studies, this frequentist method consistently and clearly indicates a large difference in mean ages. My confidence in the observed difference in average ages between the two groups is increased by the convergence of results obtained using frequentist and Bayesian approaches.
The investigation of average age differences reveals a complex terrain when comparing the Bayesian and frequentist approaches. Frequentist methods give clear interpretability, while Bayesian methods allow flexibility and measurement of uncertainty. The tension seen in the Bayesian t-test highlights how crucial it is to carefully specify the model and take previous data into account. Future analyses can gain from a comprehensive strategy that combines frequentist and Bayesian methodology’s advantages.