Bayesian Full Information Structural Analysis: with an by J.-A. Morales (auth.)

By J.-A. Morales (auth.)

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1. 2. 1J corresponding to the zero-elements drop- out. In order to lighten an already. alas, heavy notation. 1 and 6. containing J only the non-restricted elements from the vectors 6. 1. J will denote the and ~. dimensional vectors 1. J containing the unrestricted parameters t and consequently He:'. J becomes a matrix of dimension ~. ~. 1. x ~. J with ~. 1. and ~. < J- m+n-l. With this new convention the matrices (~-~O) and Mo have m m m dimension m x E ~. and I~. x E ~. respectively. i=l 1. i=l 1.

L. - 2. 1. we have the following inequality : where A~ is the smallest root of the L ~. x L ~. matrix • l. l. • l. l. MO. 38. ) J ~{(0_00)'G~(o_00)+trL-1SO}dL-l , ~>O with 0 and 00 of dimension L L ~. il. x L • l. ~. il. ~. l. 39. ) IG~12 1 t(O-OO)IG~(O-OO). %(1I-1I0)(1I-1I0) 'I 2 do « R Integrating explicitely in 0 in the second line of 2,41. 42. can be written as m 1 .. Il2 since l } { Imr-l •. IT 0 11 i=l 1 (0 11 ) 1 !.. 1. i 1 a 1 -2 1 1 1 11r- I exp- ~rr- SO 1 .. ) ax u· i 1 .. I. 43. is the kernel of a proper Wishart-density function, say fw(r-llsO, a-~ )1i+(m+I) provided that a > max )1.

J. ] = E -1 i=l J. 36. is that e be greater than a' +m+n-l. 36. is insured 'Then = 0, and the existence of the integral in e greater than a'+m+n-l. - 50- If the monomial S. contains two or more parameters we can prove . i i -1 a that E[Si] is bounded by a polynom~al ~n [a ] of degree :2' . ~ We prove this for the case of two parameters. The general case is proved by repeating the same argument. Consider a monomial S. of the following form ~ p,~ > 1 p+q < at Then taking expectations and using Schwarz' 1 E[8Ik 8{1] < ine~uality we have 1 E2[8~~]E2[8~i] = The expressions under brackets are both polynomials in (aii)-l of degree p and q respectively.

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