By Ishiguro M., Sakamoto Y.
A Bayesian technique for the chance density estimation is proposed. The method relies at the multinomial logit differences of the parameters of a finely segmented histogram version. The smoothness of the envisioned density is assured through the advent of a previous distribution of the parameters. The estimates of the parameters are outlined because the mode of the posterior distribution. The previous 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 process is constructed lower than the idea that the density is outlined on a bounded period. The dealing with of the final case the place the help of the density functionality isn't inevitably bounded can be mentioned. the sensible usefulness of the approach is validated by way of numerical examples.
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Additional info for A Bayesian Approach to the Probability Density Estimation
Studden (1990) considers n observations x1 , . . , xn from a mixture of geometric distributions, 1 x∼ θx (1 − θ) dG(θ), 0 x taking its values in IN and the probability distribution G being unknown. In this setting, G can be represented by the sequence of its noncentral moments c1 , c2 , . .. The likelihood function is then derived from P (X = k) = ck −ck+1 . Studden (1990) shows that, although the ci are constrained by an inﬁnite number of inequalities (starting with c1 > c2 > c21 ), it is possible to derive (algebraically) independent functions of the ci ’s, p1 , p2 , .
Xn ) is then (θ|x1 , . . , xn ) = f (x1 |θ)f (x2 |x1 , θ) . . f (xn |x1 , . . , xn−1 , θ)IIAn (x1 , . . , xn ), thus depends only on τ through the sample x1 , . . , xn . This implies the following principle. Stopping Rule Principle If a sequence of experiments, E1 , E2 , . , is directed by a stopping rule, τ , which indicates when the experiments should stop, inference about θ must depend on τ only through the resulting sample. 4 illustrates the case where two diﬀerent stopping rules lead to the same sample: either the sample size is ﬁxed to be 12, or the experiment is stopped when 9 viewers have been interviewed.
This principle seems diﬃcult to reject when the selected experiment is known, as shown by the following example. 5, or through a less precise but always available machine, which gives x2 ∼ N (θ, 10). The machine being selected at random, depending on the availability of the more precise machine, the inference on θ when it has been selected should not depend on the fact that the alternative machine could have been selected. 20). The equivalence result of Birnbaum (1962) is then as follows. 8 The Likelihood Principle is equivalent to the conjunction of the Suﬃciency and the Conditionality Principles.