Bayesian Statistics and Bayes Theorem
Concisely discussing one of the more important statistical philosophies in data science
If I had to guess, most of the statistical theory you’ve learned has been from a Frequentist’s prism. T-tests, z-tests, p-values, and ANOVA, just to name a few, are all from the viewpoint of Frequentists. In this blog post, I will have you consider an alternate viewpoint — the Bayesian viewpoint — as I compare the statistical structures of Bayesians versus Frequentists, and then later discuss Bayes Theorem.
Bayesians Vs. Frequentists — Philosophical Differences
Talking about their understanding of probability itself is a normal place to begin when explaining the distinctions between Bayesians and Frequentists. With Frequentists, given the same situation, including its details and assumptions, if it is replicated indefinitely, the probability of an incident is the limit of the rate of instances of that event. By comparison, Bayesians define probability as the degree of trust, or confidence, of a single occurrence happening. For certain cases, this gives a more natural explanation for rare or unusual occurrences that can not occur in the same background and conditions. The functional effects of Bayesians vs Frequentists rests on making claims concerning unspecified quantities. You make assumptions in the Bayesian system regarding uncertain variables that you are attempting to calculate.
Bayes Theorem
Bayes theorem is an essential rule of probability that relies on conditional probability and enables you to measure unknown probabilities logically. Bayes formula is represented as:
- P(A|B) = P(B|A) * P(A) / P(B)
To translate this, it is read as: the probability of ‘A’ given ‘B’ is equal to the probability of ‘B’ given ‘A’, multiplied by the probability of ‘A’, which is then divided by the probability of ‘B’. Bayes theorem is very logical, breaking down the conditional probability of ‘A given B’ in terms of the possibility that both outcomes are correct, divided by the likelihood that B is correct. This fundamental definition is taken a step further by the by representing the possibility that all outcomes are true as a conditional probability multiplied by the existing condition.
Bayesian statistics and Bayes theorem can be very dense. My objective with this post was to just go over the most important aspects of this very important area, so if you ever feel overwhelmed while studying this subject, head on over back to this post to reemphasize the most important aspects. Thank you for reading!
