statisticalrethinkin..

54 2. SMALL WORLDS AND LARGE WORLDSdirectly with the samples, because they are in many ways more convenient than having theposterior directly. And so that’s where we turn in the next chapter.2.5. Summaryis chapter introduced the conceptual mechanics of Bayesian data analysis. Bayesianprobabilities state the plausibilities of different possibilities, whether those possibilities aredata or parameters. Plausibilities are updated in light of observations, a process known asBayesian updating.A Bayesian model is a composite of a likelihood, a choice of parameters, and a prior.e likelihood provides the plausibility of an observation (data), given a fixed value for theparameters. e prior provides the plausibility of each possible value of the parameters,before accounting for the data. e rules of probability tell us that the only logical way tocompute the plausibilities, aer accounting for the data, is to use Bayes’ theorem. is resultsin the posterior distribution.In practice, Bayesian models are fit to data using numerical techniques, like grid approximation,quadratic approximation, and Markov chain Monte Carlo. Each method imposesdifferent tradeoffs.Easy.2.6. Practice2.6.1. e1. Which of the expressions below correspond to the statement: the probability ofrain on Monday?(1) Pr(rain)(2) Pr(rain|Monday)(3) Pr(Monday|rain)(4) Pr(rain, Monday)/ Pr(Monday)2.6.2. e2. Which of the following statements corresponds to the expression: Pr(Monday|rain)?(1) e probability of rain on Monday.(2) e probability of rain, given that it is Monday.(3) e probability that it is Monday, given that it is raining.(4) e probability that it is Monday and that it is raining.2.6.3. e3. Which of the expressions below correspond to the statement: the probability thatit is Monday, given that it is raining?(1) Pr(Monday|rain)(2) Pr(rain|Monday)(3) Pr(rain|Monday) Pr(Monday)(4) Pr(rain|Monday) Pr(Monday)/ Pr(rain)(5) Pr(Monday|rain) Pr(rain)/ Pr(Monday)2.6.4. e4. e Bayesian statistician Bruno de Finetti (1906–1985) began his book on probabilitytheory with the declaration: “PROBABILITY DOES NOT EXIST.” (Capitals in theoriginal.) What he meant is that probability is a device for describing uncertainty from theperspective of an observer with limited knowledge; it has no objective reality. Discuss theglobe tossing example from the chapter, in light of this statement. What does it mean to say“the probability of water is 0.7”?