Slide 1This is the last video in the chapter. We've spent four videos inside the Frequentist approach, and we ended the last one at its limit: a confidence interval still can't tell you the probability that your hypothesis is correct. The Bayesian approach can. It starts somewhere that might surprise you — with a rapid test.

Slide 2Imagine it's early in the pandemic. COVID is still rare in your community. Maybe 1 in 100 people are infected. You take a rapid test and it comes back positive. The test is accurate about 95% of the time. What's the chance you actually have COVID?

Slide 3Most people say 95%. But that's not quite right. When the disease is rare, most positive tests are actually false positives. Think about it. If you test 1,000 people and only 10 are truly infected, even a highly accurate test will flag some of the 990 healthy people by mistake. In this scenario, your actual chance of being infected might be closer to 15 or 20%, not 95%.

Slide 4Here are those two worlds side by side, a thousand people in each. On the left, COVID is rare. 1 in 100 infected, so 10 of these 1,000 people. Now imagine a month later. The virus is surging. 1 in 10 people are infected, which is 100 of them. Same test, same positive result. On the left, your chance of actually being infected if testing positive is 15 or 20%. On the right, it jumps to around 70%.

Slide 5What changed? Not the test. Not your result. Just what was true before you took the test — the prior probability. When COVID was rare, a positive result was likely to be wrong. When COVID was common, the same positive result was likely to be right. This is Bayesian reasoning in a nutshell: your conclusion depends on what you believed before you saw the evidence.

Slide 6The COVID example involves a yes or no question. Am I infected? But the same logic applies when we analyze data from a clinical trial. Instead of asking do I have COVID, we're asking what is the treatment effect. Instead of a test result, we have trial data. And instead of disease prevalence, our prior belief is about what effect sizes are plausible before we see the data.

Slide 7In the Frequentist approach we covered earlier, there's no formal role for prior beliefs. You collect data, calculate a p-value, and ask how surprising these data would be if the null hypothesis were true. Your prior knowledge might inform your study design, but it doesn't enter the statistical analysis itself.

Slide 8The Bayesian approach is different. It explicitly combines three things. The prior: what you believed about the treatment effect before collecting data. This might be a broad, open-minded prior, or a more skeptical one. The likelihood: how well different effect sizes explain the data you actually observed. This is similar to what Frequentist methods calculate. It's the information in your data. And the posterior: your updated belief after seeing the data, combining your prior belief with the evidence, weighted by how informative each is.

Slide 9The key insight is that the same data can lead to different conclusions depending on your prior. If you start skeptical and see modest evidence, you might remain unconvinced. If you start open-minded, the same evidence might shift your beliefs substantially. This isn't a bug, it's a feature. It formalizes something we do naturally: extraordinary claims require extraordinary evidence.

Slide 10Let's return to the depression trial to see what this looks like in practice. To run a Bayesian analysis, we first need to specify a prior. What we believe about the treatment effect before seeing the trial data. I used a weakly informative prior: a normal distribution centered at zero with a standard deviation of 10. What does this mean in plain terms? It says: before seeing the data, I think the most likely effect is zero, no difference between groups, but I'm open to effects anywhere from about negative 20 to positive 20 on the depression scale. Effects larger than that are almost impossible.

Slide 11This is a common choice when you want to be open-minded but not absurd. A prior centered at zero with wide spread lets the data dominate the analysis, while ruling out implausible extremes, like an improvement of 50 points on a scale that runs to 63. If we had strong prior evidence from previous trials, we might center the prior elsewhere.

Slide 12This figure compares the Frequentist 95% confidence interval reported in the trial to the Bayesian 95% credible interval, using this prior and the same trial data.

Slide 13The numbers are nearly identical. Both approaches estimate an effect around negative 7.6 points, with similar ranges of uncertainty. But the interpretation is fundamentally different. The confidence interval on the left is a Frequentist statement about long-run procedures. If we repeated this trial many times and calculated intervals each time, 95% of them would contain the true effect. It says nothing about this specific interval. Maybe the true effect is in there, maybe it isn't. We can't assign a probability. The credible interval on the right is a direct probability statement. Given the data and our prior beliefs, there is a 95% probability that the true treatment effect falls in this interval. We can make claims about this specific interval because we're describing our state of knowledge.

Slide 14Even more useful: with a posterior distribution, we can answer questions that Frequentist methods cannot. What's the probability that the effect is larger than negative 5 points? About 87%. What's the probability of any effect at all, less than zero? Over 99%. These are direct answers to the questions we actually care about.

Slide 15So if Bayesian methods give us what we want, the probability of hypotheses given data, why doesn't everyone use them? Partly, it's historical. Frequentist methods dominated twentieth-century statistics and became embedded in training, software, and journal expectations. The computational demands of Bayesian methods were also prohibitive until recently. But there are also genuine tensions. Critics worry that priors introduce subjectivity. Two researchers with different priors could reach different conclusions from the same data.

Slide 16Bayesians counter that Frequentist methods have hidden subjective choices too. Why an alpha of 0.05? Why this particular test? And that making prior assumptions explicit is more honest than pretending they don't exist. In practice, when sample sizes are large and priors are weak, Frequentist and Bayesian answers often converge, as they did in this trial. The differences matter most when data are sparse, priors are strong, or you need to make direct probability statements about hypotheses.

Slide 17Statistical inference is hard. Don't be discouraged if these concepts require multiple readings and practice to internalize. Even experienced researchers misinterpret p-values and confidence intervals. Here's what I hope you take away. Report effect sizes, not just p-values. A p-value tells you whether a result is surprising under the null hypothesis, nothing more. Understand what your intervals mean. A 95% confidence interval is a statement about a procedure. A 95% credible interval is a statement about that specific interval, but it depends on your prior. Neither is right; they answer different questions. And be honest about uncertainty. No statistical test proves a hypothesis true or false.

Slide 18Think back to where we started: a vitamin D trial where a 30% reduction in cancer risk was dismissed because the p-value was 0.06. The authors concluded no effect when the data were genuinely ambiguous. That's a threat to statistical conclusion validity, drawing the wrong conclusion from the evidence. But even if we get inference right, even if we correctly conclude the supplement probably reduces cancer risk, we still haven't answered whether it causes that reduction, or whether the effect would hold in other populations. Those are questions of internal and external validity, and they're where we turn next.

Slide 19One last thing. These five videos carry the spine of the chapter, but the chapter carries more. Go read it for the parts we left on the page. Why controlling your false positive rate at 5% does not mean that 5% of significant findings are wrong. Where the smooth bell curve in your statistics textbook comes from, and how it relates to the simulation we ran. And what an improvement of 7.57 points on the depression scale actually means clinically, including what it cost per point. The chapter has all of it, with the references.