1. What is the Laplace functional of NBP(r, l) on E=(1,)?
The Laplace functional of NBP(r, l) on E=(1,) is given by 1 + 1 (1 - e -f (x) )l(dx) -r. This functional is derived from the negative binomial process (NBP) and represents the probability measure of the process. It is a key component in understanding the behavior and properties of the NBP. The NBP is a generalization of the Poisson-Kingman distribution and is defined as a random discrete probability measure. The Laplace functional helps in analyzing the distribution and characteristics of the NBP, providing insights into its statistical properties and applications in various fields such as queueing theory, reliability analysis, and stochastic processes.
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2. What is the negative binomial process in engineering literature?
In engineering and computer science literature, the negative binomial process can have different definitions compared to the one defined in Gregoire (1984). For example, the negative binomial process defined in Zhou and Carin (2015) is different from the one defined in Zhou et al. (2012) and Broderick et al. (2015). These variations in definitions can lead to confusion, hence the need to clarify the concepts to avoid misunderstandings. Each definition may have specific applications and interpretations in the field of engineering and computer science.
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3. What is the extended Dirichlet process?
The extended Dirichlet process is a prior distribution on the space of all probability distributions developed using the probability measure (2.4) with L as the gamma Levy measure defined in (1.6). It is a natural extension of the Dirichlet process when r = 0. The process has been investigated thoroughly, showing its relation to other Poisson-Dirichlet models and its utility in data analysis. The theorem provides an efficient approximation for the extended Dirichlet process, avoiding the use of an infinite sum and using quantile functions of the Gamma(a/n, 1) distribution. It also has stochastically decreasing weights, unlike stick-breaking weights in Ipsen and Maller (2017). A similar proposal for the Dirichlet process can be found in Zarepour and Al-Labadi (2012).
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4. How to obtain posterior and predictive distribution?
To obtain the posterior and predictive distribution, we can generalize P r,L,H by assuming r is a realization of a random variable R with an arbitrary probability mass function p(r). The weights of P R,L,H can be denoted by p_i. Given observations from P R,L,H, the posterior and predictive distribution can be obtained using a recursive method from Ongaro and Cattaneo (2004). The random positions z_i are independent and identically distributed (i.i.d.) from a diffuse probability measure H. The posterior distribution of (R, p, z) can be calculated similarly to the propositions in Ongaro and Cattaneo (2004). The final theorem states that the posterior process P R,L,H |X can be represented as (P R,L,H |X) = k_i=1 g X_i d X *_i + X_{i=R} X +1 p_X = (g X_1, ..., g X_k, p_X R X +1, p_X R X +2, ...). The distribution of R X and p X are obtained using a recursive method similar to Corollaries 3 and 4 in Ongaro and Cattaneo (2004). The predictive distribution is calculated by taking the expectation of (4.1) to get Pr{X_{n+1} A|X} = k_i=1 c_X_i d X_i (A) + (1 - c_X_1 - * - * - c_X_k)H(A), where c_X_i = E(g_X_i) for i = 1, ..., k.
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