Reading 10: Bayesian Analysis II#
For the class on Wednesday, February 19th
Reading assignments#
Read the following sections of [ICVG20]:
Sec. 5.2 “Bayesian Priors”, including all the subsections (5.2.1–-5.2.4)
Sec. 5.8 “Numerical Methods for Complex Problems (MCMC)”
Lead paragraphs (text between the headings of Sec. 5.8 and Sec. 5.8.1)
Sec. 5.8.1 “Markov Chain Monte Carlo”
Sec. 5.8.2 “MCMC Algorithms”
Questions#
Submit your answer on Canvas. Due at noon, Wednesday, February 19th.
List anything from your reading that confuses you. Explain why they confuse you. If nothing confuses you, briefly summarize what you have learned from this reading assignment.
You have a coin. A flip of that coin results in heads with an unknown probability \(\theta\). You flipped the said coin 5 times, and you got heads once. Answer the following questions. Show your steps.
(a) Assuming an uniform prior, derive the posterior distribution of \(\theta\) given your flips.
(b) Compare your answer in (a) here with your answer in 2(c) from Reading 9. Are the posterior distributions in these two cases the same?
(c) In 2(a) of this reading, the data is 1 heads out of 5 flips and the prior is uniform. In 2(c) of Reading 9, the data was 0 heads out of 3 flips, and the prior was \(p(\theta) = 6 \theta (1-\theta)\). One way to interpret the prior used in 2(c) of Reading 9 is that it corresponds to a prior experiment where you flipped the coin twice and got heads once. Show that the posterior of such a prior experiment is \(p(\theta) = 6 \theta (1-\theta)\) when assuming a uniform prior.