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Monte Carlo approximation

# Monte Carlo approximation - 4 Monte Carlo approximation In...

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4 Monte Carlo approximation In the last chapter we saw examples in which a conjugate prior distribution for an unknown parameter θ led to a posterior distribution for which there were simple formulae for posterior means and variances. However, often we will want to summarize other aspects of a posterior distribution. For example, we may want to calculate Pr( θ A | y 1 , . . . , y n ) for arbitrary sets A . Alternatively, we may be interested in means and standard deviations of some function of θ , or the predictive distribution of missing or unobserved data. When comparing two or more populations we may be interested in the posterior distribution of | θ 1 - θ 2 | , θ 1 2 , or max { θ 1 , . . . , θ m } , all of which are functions of more than one parameter. Obtaining exact values for these posterior quantities can be difficult or impossible, but if we can generate random sample values of the parameters from their posterior distributions, then all of these posterior quantities of interest can be approximated to an arbitrary degree of precision using the Monte Carlo method. 4.1 The Monte Carlo method In the last chapter we obtained the following posterior distributions for birthrates of women without and with bachelor’s degrees, respectively: p ( θ 1 | 111 i =1 Y i, 1 = 217) = dgamma( θ 1 , 219 , 112) p ( θ 2 | 44 i =1 Y i, 2 = 66) = dgamma( θ 2 , 68 , 45) Additionally, we modeled θ 1 and θ 2 as conditionally independent given the data. It was claimed that Pr( θ 1 > θ 2 | Y i, 1 = 217 , Y i, 2 = 66) = 0 . 97. How was this probability calculated? From Chapter 2, we have P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/978-0-387-92407-6 4, c Springer Science+Business Media, LLC 2009

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54 4 Monte Carlo approximation Pr( θ 1 > θ 2 | y 1 , 1 , . . . , y n 2 , 2 ) = 0 θ 1 0 p ( θ 1 , θ 2 | y 1 , 1 , . . . , y n 2 , 2 ) 2 1 = 0 θ 1 0 dgamma( θ 1 , 219 , 112) × dgamma( θ 2 , 68 , 45) 2 1 = 112 219 45 68 Γ (219) Γ (68) 0 θ 1 0 θ 218 1 θ 67 2 e - 112 θ 1 - 45 θ 2 2 1 . There are a variety of ways to calculate this integral. It can be done with pencil and paper using results from calculus, and it can be calculated nu- merically in many mathematical software packages. However, the feasibility of these integration methods depends heavily on the particular details of this model, prior distribution and the probability statement that we are trying to calculate. As an alternative, in this text we will use an integration method for which the general principles and procedures remain relatively constant across a broad class of problems. The method, known as Monte Carlo approxima- tion , is based on random sampling and its implementation does not require a deep knowledge of calculus or numerical analysis. Let θ be a parameter of interest and let y 1 , . . . , y n be the numerical values of a sample from a distribution p ( y 1 , . . . , y n | θ ). Suppose we could sample some number S of independent, random θ -values from the posterior distribution p ( θ | y 1 , . . . , y n ): θ (1) , . . . , θ ( S ) i.i.d p ( θ | y 1 , . . . , y n ) .
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