Bayes In the Real World: The Metropolis-Hastings AlgorithmHere we have each individual sample and the true posterior distribution. We can see a few outliers in the samples that distort the final results, but overall things look pretty much on the money. If we were doing this analysis for practical rather than educational purposes, we'd want to ask questions of our samples:
- What's the most likely value?
- What's the expected value?
- What's the 95% highest density interval (HDI)—the top 95% most likely values from a random sample?1
These chains would give quality answers to these questions.
- Note the wording: an HDI is not a confidence interval. A credible interval is the Bayesian version of a confidence interval—an interval that contains 95% of the area. Despite the name, an HDI doesn't have to be contiguous: if our distribution had two peaks, an HDI can split to cover both, whereas a confidence interval will misrepresent our beliefs.↩
Here we have each individual sample and the true posterior distribution. We can see a few outliers in the samples that distort the final results, but overall things look pretty much on the money. If we were doing this analysis for practical rather than educational purposes, we'd want to ask questions of our samples:
- What's the most likely value?
- What's the expected value?
- What's the 95% highest density interval (HDI)—the top 95% most likely values from a random sample?1
These chains would give quality answers to these questions.
- Note the wording: an HDI is not a confidence interval. A credible interval is the Bayesian version of a confidence interval—an interval that contains 95% of the area. Despite the name, an HDI doesn't have to be contiguous: if our distribution had two peaks, an HDI can split to cover both, whereas a confidence interval will misrepresent our beliefs.↩