By Ishiguro M., Sakamoto Y.

A Bayesian approach for the chance density estimation is proposed. The method relies at the multinomial logit alterations of the parameters of a finely segmented histogram version. The smoothness of the anticipated density is assured via the advent 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 details Criterion) is minimized.The uncomplicated method is built less than the belief that the density is outlined on a bounded period. The dealing with of the final case the place the aid of the density functionality isn't unavoidably bounded can be mentioned. the sensible usefulness of the method is tested via numerical examples.

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**Example text**

Therefore the implication (b)~(c) always holds. 7 Corollary. 8 Corollary. the following including of Kwapien which asserts of Rademacher then E is isomorphic isomor- Clearly, type p and of to an L p space. poses the problem whether a Banach lattice type p and Poisson cotype p must be isomorphic that E contains that a type and cotype 2 is necessarily If E is a Banach lattice cotype p (I~ p ~ 2 ) L p space. for the space. 2 one has to exclude a copy of ~I. The solution but we will give some results the case of this problem is open, going in this direction.

Ix K~J. uP) K~l P(d~) ~(dt) "A'A"'4 i G ~ " k,:4 | ~ k k ~l (~)x~. ~ ~'~ ~k,i p ~ ii Jl 30 Remark. The above theorem can be proved directly for an arbitrary abstract L p space without using any concrete representation of these spaces. 59). 2 Proposition. Let E be an abstract L I space. 1 holds for every finite sequence (Xk) of independent E+-valued and p-integrable random vectors. Proof. ~ The above two propositions the following definition. 3 Definition. A Banach lattice E is called a Banach lattice of Poisson type p (I~ p < ~ ) if there exists a constant C>o only depending on p such that for all finite sequences (~k) of independent Poisson distributed real random variables with parameters ~k ~ I and for all finite sequences (Xk) in E+ the following inequality holds: JU~" ~kXkJP dP ~ C (~- ~klXkl|P + ~ " ~kXk|P).

The mapping ~ :S--~ ~R+ is given by ~(IA):=~4(A) for AC2X . 2. The following example may indicate that it can be useful to have the above existence theorem in its stated general form. ,~n) 39 of simple measures with disjoint supports. Then the proposition shows that to any semigroup (~t)tCT of probability measures on T there exists a process ~= (~s)s~S with the properties stated in the proposition. Here S is the subset of M+(X,Z+) consisting of all simple measures. 2 Corollary. Let ( X ~ , ~ ) be a measure space and put ~o := { S E Z : ~ ( S ) , ~ ] .

### A Bayesian Approach to the Probability Density Estimation by Ishiguro M., Sakamoto Y.

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