Chapter 5 · Section 1 practice

Bayesian Foundations and Risk Estimates

0 of 8 answered

These questions review the examples in the companion slides. Choose an answer, then check the explanation.

Question 1

After seeing 1 258 daily returns, which object lets you say “P(reward-to-t-scale > 1) = 0.99”?

  • The likelihood \(p(X \mid \theta)\)
  • The prior \(p(\theta)\)
  • The posterior \(p(\theta \mid X)\)
  • The p-value of a t-test on the mean

Question 2

Which choice of \(\nu\) produces the fattest tails?

  • \(\nu = 100\) — essentially normal
  • \(\nu = 3\)
  • \(\nu = 30\)
  • Tail weight does not depend on \(\nu\)

Question 3

What does the prior \(\nu = 2 + \text{Exp}(1/29)\) force?

  • Only that \(\nu > 2\); the data decide the rest
  • That \(\nu \approx 31\)
  • That the tails are normal
  • That \(\nu\) is an integer

Question 4

A 400-point grid per parameter and a 3-parameter model (\(\mu, \sigma, \nu\)). How many likelihood evaluations?

  • 1 200
  • 160 000
  • 4 800
  • 64 000 000

Question 5

\(p^\star = 0.3\,p_t\). What does Metropolis do?

  • Reject — the proposal is worse
  • Accept with probability 0.3
  • Accept — proposals are always accepted
  • Accept only if 0.3 is larger than the prior

Question 6

What does a healthy trace look like?

  • A flat, fuzzy band with no trend
  • A smooth upward drift
  • Long horizontal segments with occasional jumps
  • A single spike

Question 7

The posterior mean reward-to-t-scale is 2.38 against a conventional Sharpe estimate of 0.77. The main reason?

  • The prior on \(\mu\) pushes the mean up
  • MCMC is biased upward
  • The t-scale \(\sigma\) is smaller than the sample sd because the tails are absorbed by \(\nu\)
  • The classical formula forgets \(\sqrt{252}\)

Question 8

For a standard-deviation Sharpe, which interval supports a 95% posterior probability statement conditional on its model and prior?

  • The confidence interval [-0.11, 1.65]
  • The sd-basis credible interval [0.58, 1.85]
  • Both — they mean the same thing
  • Neither — probability statements about parameters are impossible