Follow-up questions and answers to the Exploring PDFs section.

Make sure to look at 📥 Exploring PDFs first!

Confidence intervals

Bayesian confidence intervals: how are they defined?

Answer

The Bayesian confidence or credible interval for a parameter with a one-dimenional PDF is defined for a given percentage, call it \(\beta\%\). The interval is such that integrating over it one gets \(\beta\%\) of the toal area under the PDF. This may not be uniquely defined if the PDF is multimodal or skewed.

Point estimates

Various “point estimates” were introduced (mean, mode, median); which is “best” to use?

Answer

A point estimate is a one-number summary of a distribution. Which one is best to use depends on the application. Sometimes one wants to know the most likely case: then use the mode. Sometimes one really wants the average: then use the mean. Sometimes, for an asymmetric distribution, the median is the best estimate. For a symmetric, unimodal PDF, the three point estimates are the same. Note that to a Bayesian, a point estimate is only a rough approximation to the information in the full distribution.

Characteristics of PDFs

What are the characteristics (e.g., symmetry, heavy tails, …) of different pdfs: normal, beta, student t, \(\chi^2\), \ldots

Answer

Check these yourself! Note that the answers will often depend on the parameter values for the distribution. A Student t distribution may have “heavy tails” (meaning more probability in the tails than a Gaussian would have) for some parameters but for others it approaches a normal distribution (so by construction no heavy tails).

Sampling

What does sampling mean?

Answer

To sample a given distribution is to draw values with probabilities given by the distribution. In Bayesian inference one is most often sampling a posterior distribution.

Verifying sampling

How do you know if a distribution is correctly sampled?

Answer

One way is to look at the (normalized)histogram. If correctly sampled, this should approximate the distribution and the approximation should improve with more samples.