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ssrs 2012 barcode font Bayesian Methods in Software
CHAPTER 11 Bayesian Methods Scanning QR Code In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Making QRCode In None Using Barcode maker for Software Control to generate, create QR Code ISO/IEC18004 image in Software applications. When a b 1 in Theorem 115, the posterior mode estimate gpost of u reduces to the maximum likelihood estimate x>n. This was also pointed out in Example 11.13 but the result is obviously not true for general a and b. But, regardless of the values of a and b, when the sample size is large enough, both mpost and gpost will be close to the sample proportion x>n. Furthermore, for all n, mpost is a convex combination of the prior mean of u and the sample proportion. (See Problem 11.38.) Recognizing QR Code In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Paint QR Code In Visual C#.NET Using Barcode printer for .NET framework Control to generate, create QR Code 2d barcode image in .NET framework applications. EXAMPLE 11.14 Suppose that a random sample of size n is drawn from a normal distribution with unknown mean u and variance 1. Also suppose that the prior distribution of u is normal with mean 0 and variance 1. From Theorem 113, we see that the posterior distribution of u is normal with mean nx >(1 n). # Clearly, the posterior mean, median, and mode are identical here and would therefore lead to the same point estimate, nx >(1 n), of u. It was shown in Problem 6.25, page 206, that the maximum likelihood estimate for u in this case is the # sample mean x, which is known to be unbiased (Theorem 51). On the other hand, the Bayesian estimates derived here # are biased, although they are asymptotically unbiased. Quick Response Code Maker In .NET Framework Using Barcode creation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. QR Drawer In Visual Studio .NET Using Barcode printer for VS .NET Control to generate, create QR image in .NET framework applications. A general result along these lines follows easily from Theorem 113 and is as follows. Theorem 116 Suppose that a random sample of size n is drawn from a normal distribution with unknown mean u and known variance s2. Also suppose that the prior distribution of u is normal with mean m and variance y2. Then the posterior mean, median, and mode all provide the same estimate of u, namely (s2m ny2 x)>(s2 ny2), where x is the sample mean. # # Generating QR Code In Visual Basic .NET Using Barcode drawer for VS .NET Control to generate, create QR Code image in VS .NET applications. Make EAN13 Supplement 5 In None Using Barcode creation for Software Control to generate, create EAN13 image in Software applications. As we saw in the binomial case, the posterior mean estimate mpost just obtained lies between the prior mean m and the maximum likelihood estimate x of u. This may be seen by writing mpost in the form # [s2 >(s2 ny2)] m [ny2 >(s2 ny2)] x, as a convex combination of the two. We can also see from this ex# pression that for large n, mpost will be close to x and will not be appreciably influenced by the prior mean m. # An optimality property of mpost as an estimate of u directly follows from Theorem 36. Indeed, we can prove a more general result along these lines using this theorem. Suppose we are interested in estimating a function of u, say t(u). For any set of observations x from f (x Zu), if we define the statistic T(x) as the posterior expectation of t(u), namely Code39 Generator In None Using Barcode drawer for Software Control to generate, create Code 39 Full ASCII image in Software applications. Create DataMatrix In None Using Barcode creator for Software Control to generate, create Data Matrix ECC200 image in Software applications. T(x) then it follows from Theorem 36 that
Create Barcode In None Using Barcode creator for Software Control to generate, create barcode image in Software applications. Barcode Creation In None Using Barcode encoder for Software Control to generate, create bar code image in Software applications. E(t(u)u x) USD8 Drawer In None Using Barcode generation for Software Control to generate, create USD  8 image in Software applications. Recognizing Bar Code In C# Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications. 3 t(u)p(uux) du
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T(x). In other words, T(x) satisfies the property E[(t(u) T(x))2 u x] min E[(t(u) a a(x))2 u x] for each x (12) since T(x) is the mean of t(u) with respect to the posterior density p(u u x). In the general theory of Bayesian estimation, we typically start with a loss function L(t(u), a) that measures the distance between the parameter and an estimate. We then seek an estimator, say d*(X), with the property that E[L(t(u), d*(x))u x] min E[L(t(u), a(x))u x] for each value x of X a (13) where the expectation is over the parameter space endowed with the posterior density. An estimator satisfying equation (13) is called a Bayes estimator of t(u) with respect to the loss function L(t(u), a). The following theorem then is just a restatement of (12): Theorem 117 The mean of t(u) with respect to the posterior distribution p(u u X) is the Bayes estimator of t(u) for the squared error loss function L(u, a) (u a)2. Another common loss function is the absolute error loss function L(u, a) u u au. It is shown in Problem 11.100 that the median of the posterior density is the Bayes estimator for this loss function.

