Posterior intervals and decisions

Saying how sure you are

The posterior is a whole spread of beliefs, so you can answer questions with probabilities. A $95\%$ credible interval is a range that holds the true value with $95\%$ probability, given your prior and the data. This is the plain sentence people wrongly attach to ordinary confidence intervals. For the normal posterior $N(9.66\%,\,3.12\%)$ the interval is the mean plus or minus $1.96$ standard deviations: $[3.5\%,\ 15.8\%]$. For the win rate $\mathrm{Beta}(19,11)$ it is $[0.46,\ 0.79]$, and it is not symmetric around the mean.

You can also ask for the probability of exactly what you care about, such as $P(\mu>c)$. A decision needs one more thing: what it costs to be wrong. Say the strategy pays off $+2$ units if it is good and loses $10$ units if it is not. With a posterior chance of $0.93$ that it is good, the expected result is $0.93\times2-0.07\times10=1.16$, so you trade. If the chance were $0.80$ you would get $0.8\times2-0.2\times10=-0.4$ and stay out. The best single guess depends on how you score mistakes: the posterior mean is best for squared errors and the median for absolute errors.

• Credible interval: a range holding the true value with the stated probability, given prior and data.
• Tail probabilities such as $P(\mu>c)$ answer the question you actually have.
• Decisions: average the payoff over the posterior and take the action with the best expected result.

Lab · what's graded

• The lower end15 pts
• The upper end15 pts
• A tail probability15 pts
• The expected result of a bet15 pts
• Reading a credible interval10 pts
• Match the vocabulary15 pts
• Short record, wide interval15 pts