By Ishiguro M., Sakamoto Y.

A Bayesian strategy for the chance density estimation is proposed. The strategy is predicated at the multinomial logit adjustments of the parameters of a finely segmented histogram version. The smoothness of the expected density is assured via the creation of a previous distribution of the parameters. The estimates of the parameters are outlined because the mode of the posterior distribution. The earlier distribution has a number of adjustable parameters (hyper-parameters), whose values are selected in order that ABIC (Akaike's Bayesian info Criterion) is minimized.The easy strategy is built below the idea that the density is outlined on a bounded period. The dealing with of the overall case the place the help of the density functionality isn't really unavoidably bounded can be mentioned. the sensible usefulness of the approach is proven through numerical examples.

Show description

Read or Download A Bayesian Approach to the Probability Density Estimation PDF

Similar probability books

Introduction to Probability Models (10th Edition)

Ross's vintage bestseller, creation to likelihood types, has been used largely by way of professors because the fundamental textual content for a primary undergraduate direction in utilized chance. It presents an advent to hassle-free likelihood concept and stochastic procedures, and exhibits how likelihood concept might be utilized to the research of phenomena in fields reminiscent of engineering, desktop technology, administration technology, the actual and social sciences, and operations study.

Real analysis and probability

This vintage textbook, now reissued, deals a transparent exposition of contemporary chance conception and of the interaction among the houses of metric areas and likelihood measures. the recent version has been made much more self-contained than prior to; it now encompasses a starting place of the genuine quantity process and the Stone-Weierstrass theorem on uniform approximation in algebras of services.

Additional info for A Bayesian Approach to the Probability Density Estimation

Example text

XjÀ1 ; xjþ1 ; . . ; xn Þd xj À1 ¼ pðx1 ; . . ; xk jxkþ1 ; . . ; xjÀ1 ; xjþ1 ; . . ; xn Þ ð2:117Þ In particular, the following formula (playing a prominent role in the theory of Markov processes) can be obtained ð1 pðx1 jx2 ; x3 Þpðx2 jx3 Þd x2 ð2:118Þ pðx1 jx3 Þ ¼ À1 All the considered definitions and rules remain valid for the case of a discrete random variable, with integrals being reduced to sums. Random variables, 1 ; 2 ; . . ; n are called mutually independent if events f1 < x1 g; f2 < x2 g; .

193) that pA; ðA; Þ ¼ pA ðAÞp ðÞ and thus the phase and the magnitude are independent. 191), m 1 1 cos A þ m À 2Am A m 4À ðA; Þ ¼ exp 2 2 2  2 2 pA; 2 2 3 m1 2 þ sin m 5 ! A A2 þ m2 À 2 A m cosð À 0 Þ ¼ exp À 2  2 2 2 ð2:194Þ Here tan 0 ¼ m1 2 m1 1 ð2:195Þ Further integration over the phase variable  produces the Rice distribution for the magnitude ! ð !   A A2 þ m 2  A m cosð À 0 Þ A A2 þ m 2 Am exp À exp exp À d  ¼ I 0 2  2 2 2 2 2 2 2 2 2 À ð ð ð2:196Þ 42 RANDOM VARIABLES AND THEIR DESCRIPTION Similarly, integration of pA; ðA; Þ over A produces a PDF of the phase with the following form  !

Indeed, it is shown in [9] that cumulant coefficients n of a random variable n ¼ n n=2 2 ¼ n n ð2:234Þ must satisfy certain (non-linear) inequalities. For example, skewness 3 and curtosis 4 must satisfy the condition [9] 4 À 32 þ 2 ! e. 3 2 ðÀ1; 1Þ; 4 must exceed À2. Restrictions on higher order cumulants are still an area of active research. 8 CUMULANT EQUATIONS It was shown earlier that the characteristic function Âð j uÞ can be defined if an infinite set of cumulants k ; k ¼ 1; 2; . , is given.

Download PDF sample

Rated 4.29 of 5 – based on 34 votes