**Related to**: Bayes' Theorem Illustrated, What is Bayesianism?, An Intuitive Explanation of Bayes' Theorem

*(Bayes' theorem is something Bayesians need to use more often than Frequentists do, but Bayes' theorem itself isn't Bayesian. This post is meant to be a light introduction to the difference between Bayes' theorem and Bayesian data analysis.)*

**Bayes' Theorem**

Bayes' theorem is just a way to get (e.g.) p(B|A) from p(A|B) and p(B). The classic example of Bayes' theorem is diagnostic testing. Suppose someone either has the disease (D^{+}) or does not have the disease (D^{-}) and either tests positive (T^{+}) or tests negative (T^{-}). If we knew the sensitivity P(T^{+}|D^{+}), specificity P(T^{-}|D^{-}) and disease prevalence P(D^{+}), then we could get the positive predictive value P(D^{+}|T^{+}) using Bayes' theorem:

For example, suppose we know the sensitivity=0.9, specificity=0.8 and disease prevalence is 0.01. Then,

This answer is not Bayesian or frequentist; it's just correct.

**Diagnostic testing study**

Typically we will not know P(T^{+}|D^{+}) or P(T^{-}|D^{-}). We would consider these unknown parameters. Let's denote them by Θ_{sens} and Θ_{spec}. For simplicity, let's assume we know the disease prevalence P(D^{+}) (we often have a lot of data on this).

Suppose 1000 subjects with the disease were tested, and 900 of them tested positive. Suppose 1000 disease-free subjects were tested and 200 of them tested positive. Finally, suppose 1% of the population has the disease.

**Frequentist approach**

Estimate the 2 parameters (sensitivity and specificity) using their sample values (sample proportions) and plug them in to Bayes' formula above. This results in a point estimate for P(D^{+}|T^{+}) of 0.043. A standard error or confidence interval could be obtained using the delta method or bootstrapping.

Even though Bayes' theorem was used, this is not a Bayesian approach.

**Bayesian approach**

The Bayesian approach is to specify prior distributions for all unknowns. For example, we might specify independent uniform(0,1) priors for Θ_{sens} and Θ_{spec}. However, we should expect the test to do at least as good as guessing (guessing would mean randomly selecting 1% of people and calling them T^{+}). In addition, we expect Θ_{sens}>1-Θ_{spec}. So, I might go with a Beta(4,2.5) distribution for Θ_{sens }and Beta(2.5,4) for Θ_{spec}:

Using these priors + the data yields a posterior distribution for P(D+|T+) with posterior median 0.043 and 95% credible interval (0.038, 0.049). In this case, the Bayesian and frequentist approaches have the same results (not surprising since the priors are relatively flat and there are a lot of data). However, the methodology is quite different.

**Example that illustrates benefit of Bayesian data analysis**

(example edited to focus on credible/confidence intervals)

Suppose someone shows you what looks like a fair coin (you confirm head on one side tails on the other) and makes the claim: "This coin will land with heads up 90% of the time"

Suppose the coin is flipped 5 times and lands with heads up 4 times.

**Frequentist approach**

"A 95% confidence interval for the Binomial parameter is (.38, .99) using the Agresti-Coull method." Because 0.9 is within the confidence limits, the usual conclusion would be that we do not have enough evidence to rule it out.

**Bayesian approach**

"I don't believe you. Based on experience and what I know about the laws of physics, I think it's very unlikely that your claim is accurate. I feel very confident that the probability is close to 0.5. However, I don't want to rule out something a little bit unusual (like a probability of 0.4). Thus, my prior for the probability of heads is a Beta(30,30) distribution."

After seeing the data, we update our belief about the binomial parameter. The 95% credible interval for it is (0.40, 0.64). Thus, a value of 0.9 is still considered extremely unlikely.

This illustrates the idea that, from a Bayesian perspective, implausible claims require more evidence than plausible claims. Frequentists have no formal way of including that type of prior information.