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Bayesian Methods
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Subjective Probability
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The statistical methods developed thus far in this book are based entirely on the classical and frequency approaches to probability (see page 5). Bayesian methods, on the other hand, rely also on a third the so-called subjective or personal view of probability. Central to Bayesian methods is the process of assigning probabilities to parameters, hypotheses, and models and updating these probabilities on the basis of observed data. For example, Bayesians do not treat the mean u of a normal population as an unknown constant; they regard it as the realized value of a random variable, say , with a probability density function over the real line. Similarly, the hypothesis that a coin is fair may be assigned a probability of 0.3 of being true, reflecting our degree of belief in the coin being fair. In the Bayesian approach, the property of randomness thus appertains to hypotheses, models, and fixed quantities such as parameters as well as to variable and observable quantities such as conventional random variables. Probabilities that describe the extent of our knowledge and ignorance of such nonvariable entities are usually referred to as subjective probabilities and are usually determined using one s intuition and past experience, prior to and independently of any current or future observations. In this book, we shall not discuss the controversial yet pivotal issue of the meaning and measurement of subjective probabilities. Rather, our focus will be on how prior probabilities are utilized in the Bayesian treatment of some of the statistical problems covered earlier.
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EXAMPLE 11.1 Statements involving classical probabilities: (a) the chances of rolling a 3 or a 5 with a fair die are one in three; (b) the probability of picking a red chip out of a box containing two red and three green chips is two in five. Examples of the frequency approach to probability: (a) based on official statistics, the chances are practically zero that specific person in the U.S. will die from food poisoning next year; (b) I toss a coin 100 times and estimate the probability of a head coming up to be 37>100 0.37. Statements involving subjective probabilities: (a) he is 80% sure that he will get an A in this course; (b) I believe the chances are only 1 in 10 that there is life on Mars; (c) the mean of this Poisson distribution is equally likely to be 1, 1.5, or 2.
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Prior and Posterior Distributions
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The following example is helpful for introducing some of the common terminology of Bayesian statistics.
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EXAMPLE 11.2 A box contains two fair coins and a biased coin with probability for heads P(H) 0.2. A coin is chosen at random from the box and tossed three times. If two heads and a tail are obtained, what is the probability of the event F, that the chosen coin is fair, and what is the probability of the event B, that the coin is biased
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Let D denote the event (data) that two heads and a tail are obtained in three tosses. The conditional probability P(D u F ) of observing the data under the hypothesis that a fair coin is tossed is a binomial probability and may be obtained from (1), (see 4). The conditional probability P(D u B) of observing D when a biased coin is tossed may be obtained similarly. Bayes theorem (page 8) then gives us P(D u F )P(F ) P(D u F )P(F ) P(D u B)P(B) P(F u D) < 0.11. 2 [3(0.5)3] 3 2 [3(0.5)3] 3 1 [3(0.2)2(0.8)] 3 250 < 0.89 282
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P(F u D)
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Also, P(B u D)
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CHAPTER 11 Bayesian Methods
In the Bayesian context, the unconditional probability P(F) in the preceding example is usually referred to as the prior probability (before any observations are collected) of the hypothesis F, that a fair coin was tossed, and the conditional probability P(F uD) is called and the posterior probability of the hypothesis F (after the fact that D was observed). Analogously, P(B), and P(B u D) are the respective prior and posterior probabilities that the biased coin was tossed. The prior probabilities used here are classical probabilities. The following example involves a simple modification of Example 11.2 that necessitates an extension of the concept of randomness and brings into play the notion of a subjective probability.
EXAMPLE 11.3 A box contains an unknown number of fair coins and biased coins (with P(H) 0.2 each). A coin is chosen at random from the box and tossed three times. If two heads and a tail are obtained, what is the probability that the chosen coin is biased In Example 11.2, the prior probability P(F ) for choosing a fair coin could be determined using combinatorial reasoning. Since the proportion of fair coins in the box is now unknown, we cannot access P(F ) as a classical probability without resorting to repeated independent drawings from the box and approximating it as a frequency ratio. We cannot therefore apply Bayes theorem to determine the posterior probability for F.
Bayesians, nonetheless, would provide a solution to this by first positing that the unknown prior probability P(F) is a random quantity, say , by virtue of our uncertainty as to its exact value and then reasoning that it is possible to arrive at a probability or density function p(u) for that reflects our degree of belief in various propositions concerning P(F ). For example, one could argue that in the absence of any evidence to the contrary before the coin is tossed, it is reasonable to assume that the box contains an equal number of fair and biased coins. Since P(H) 0.2 for a biased coin and 0.5 for a fair coin, the unknown parameter then would have the subjective prior probability function shown in Table 11-1. Table 11-1 u p(u) 0.2 1>2 0.5 1>2
Prior distributions that give equal weight to all possible values of a parameter are examples of diffuse, vague, or noninformative priors which are often recommended when virtually no prior information about the parameter is available. When a parameter can take on any value in a finite interval, the diffuse prior would usually be the uniform density on that interval. We will also encounter situations where uniform prior densities over the entire real line are used; such densities will be called improper since the total area under them is infinite. Starting from the prior probability function in Table 11-1, the posterior probability function for after observing D (two heads and a tail in three tosses), p(u u D), may be obtained using Bayes theorem as in Example 11.2, and is given in Table 11-2 (see Problem 11.3).