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ssrs 2012 barcode font and (ii) 3 p(u) du ` in Software
and (ii) 3 p(u) du ` Reading QR Code ISO/IEC18004 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Encoding QR Code JIS X 0510 In None Using Barcode generation for Software Control to generate, create QRCode image in Software applications. 1 (see page 37). Prior densities that satisfy the first condition but violate the second due
Decode QRCode In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Encode QR Code In Visual C# Using Barcode maker for .NET framework Control to generate, create QR Code image in Visual Studio .NET applications. to the integral being divergent have, however, been employed within the Bayesian framework and are referred to as improper priors. They often arise as natural choices for representing vague prior information about parameters with infinite range. For example, when sampling from a normal population with known mean, say 0, but unknown variance u, we 1 , u 0. Given a sample of observations may assume that the prior density for u is given by p(u) u Creating QR Code In .NET Framework Using Barcode encoder for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Create QR Code In Visual Studio .NET Using Barcode creation for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications. CHAPTER 11 Bayesian Methods
Painting QR Code 2d Barcode In Visual Basic .NET Using Barcode generator for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. UCC  12 Drawer In None Using Barcode creation for Software Control to generate, create GTIN  128 image in Software applications. x (x1, x2, c, xn), if we overlook the fact that the prior is improper and apply formula (1), we get the posterior density a x2 i Barcode Drawer In None Using Barcode drawer for Software Control to generate, create barcode image in Software applications. EAN / UCC  13 Generator In None Using Barcode creation for Software Control to generate, create EAN 13 image in Software applications. 1 p(uu x) ~ n>2 exp u
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Barcode Creator In Java Using Barcode drawer for Eclipse BIRT Control to generate, create bar code image in BIRT applications. UPC Code Encoder In VB.NET Using Barcode creator for VS .NET Control to generate, create UPCA Supplement 5 image in Visual Studio .NET applications. (10) Drawing GS1128 In Java Using Barcode encoder for BIRT Control to generate, create EAN128 image in BIRT applications. Data Matrix Creation In VB.NET Using Barcode generation for Visual Studio .NET Control to generate, create Data Matrix image in .NET framework applications. This is a proper density, known as an inverse gamma, with parameters a n>2 and b g i x2 >2 (see Problem i 11.99). We have thus arrived at a proper posterior density starting with an improper prior. Indeed, this will be true in all of the situations with improper priors that we encounter here, although this is not always the case. EXAMPLE 11.9 Suppose that X is binomial with known n and unknown success probability u. The prior density for 1 u given by p(u) , 0 u 1 is improper and is known as Haldane s prior. Let us overlook the fact that p(u) u(1 u) is improper, and proceed formally to derive the posterior density p(u u x) corresponding to an observed value x of X: p(u u x) f (x u u)p(u) 3 f (x u u)p(u)du
ux(1 u(1 u)n u) ux 1(1 u)n x B(x, n x) We see that the posterior is a proper beta density with parameters x and n x.
EXAMPLE 11.10 Suppose X is normally distributed with unknown mean u and known variance s2. An improper prior u ` . This density may be thought of as representing prior distribution for u in this case is given by p(u) 1, ` ignorance in that intervals of the same length have the same weight regardless of their location on the real line. Given the observation vector x (x1, x2, c, xn), the posterior distribution of u under this prior is given by a (xi p(u u x) ~ f (xu u)p(u) ~ exp
1 ~ exp e
n(u x)2 # f 2s2
which is normal with mean x and variance s2 >n. # Conjugate Prior Distributions
Note that Theorems 111, 112, and 113 share an important characteristic in that the prior and posterior densities in each belong to the same family of distributions. Whenever this happens, we say that the family of prior distributions used is conjugate (or closed) with respect to the population density f (x uu). Thus the beta family is conjugate with respect to the binomial distribution (Theorem 111), the gamma family is conjugate with respect to the Poisson distribution (Theorem 112), and the normal family is conjugate with respect to the normal distribution with known variance (Theorem 113). Since p(u u x, y) ~ f ( y u u)p(uu x) whenever x and y are two independent samples from f (x uu), conjugate families make it easier to update prior densities in a sequential manner by just changing the parameters of the family (see Example 11.11). Conjugate families are thus desirable in Bayesian analysis and they exist for most of the commonly encountered distributions. In practice, however, prior distributions are to be chosen on the basis of how well they represent one s prior knowledge and beliefs rather than on mathematical convenience. If, however, a conjugate prior distribution closely approximates an appropriate but otherwise unwieldy prior distribution, then the former naturally is a prudent choice. We now show that the gamma family is conjugate with respect to the exponential distribution. Suppose that X has the exponential density, f (x u u) ue ux, x 0, with unknown u, and that the prior density of u is gamma with parameters a and b. The posterior density of u is then given by p(uu x) ~ f (x u u) p(u) une

