Identifying Speculative Bubbles with an Infinite Hidden Markov Model

ADirichlet Process and Stick Breaking ProcessA.1 Dirichlet ProcessBefore introducing the Dirichlet process, the definition of the Dirichlet distribution is thefollowing:Definition The Dirichlet distribution is denoted by Dir(α), where α is a K-dimensionalvector of positive values. Each sample x from Dir(α) is a K-dimensional vector with x i ∈ (0, 1)∑and K x i = 1. The probability density function isi=1p(x | α) =∑Γ( K α i ) K∏i=1K∏Γ(α i )i=1i=1x α i−1iA special case is the Beta distribution, where K = 2.∑Define α 0 = K α i and X i as the ith element of the random vector X from a Dirichleti=1distribution Dir(α). The random variable X i has mean α iα 0and variance α i(α 0 −α i )α 2 0 (α 0+1) . Hence,we can further decompose α into two parts: a shape parameter G 0 = ( α 1α 0, · · · , α Kα0) and aconcentration parameter α 0 . The shape parameter G 0 represents the center of the randomvector X and the concentration parameter α 0 controls how close X is to G 0 .The Dirichlet distribution is conjugate to the multi-nominal distribution in the followingsense: ifX ∼ Dir(α),β = (n 1 , . . . , n K ) | X ∼ Mult(X),25