Prior Processes and Their Applications by Eswar G. Phadia

Author:Eswar G. Phadia
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

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where m k is the number of customers among n having tried dish k and ω k stands for ‘tastes dish k’. It’s interpretation in the terminology of Indian Buffet process is as follows. Suppose B 0 is continuous and c is constant such that λ=B 0(Ω) is finite. Since Z 1∼BeP(B 0) and B 0 is continuous, Z 1 is a Poisson process (B 0), and the total number of features of Z 1 is Z 1(Ω)∼P(λ). That is the first customer will taste P(λ) number of dishes. For the (n+1)-th customer, Z n+1 is sum of two components: U the number of dishes already tasted by n customers, and V the number of new dishes he will taste. and . U will have mass at ω k i.e. he will taste dish k already tried by previous customers with probability , k=1,…,K, and will taste additionally number of new dishes.

Here the underlying Dirichlet/multinomial structure for the CRP is replaced by beta/Bernoulli structure. For an application to document classification problem, their paper should be consulted.

Stick-Breaking Construction of IBP

To sample a binary matrix from the distribution of Z we need μ k ’s (similar to p i ’s in the Sethuraman representation of the Dirichlet process). But since we do not care for the ordering of columns, it is sufficient to generate ordered μ k ’s. Then these ordered μ k ’s are given in terms of θ k ’s with each θ k ∼Be(α,1). Let μ (1)>μ (2)>…>μ (K) be a decreasing reordering of μ 1,μ 2,…,μ K , where a prior is placed on each μ k . As K→∞, Teh et al. (2007) construct a stick-breaking representation of the Indian buffet process as follows. Let

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