Identifying Speculative Bubbles with an Infinite Hidden Markov Model

Suppose there are n observations, a = (a 1 , · · · , a n ), drawn from the distribution F . Usen∑δ ai (A j ) toi=1represent the number of a i in set A j , where A 1 , · · · , A K is a measurable partition of thesample space Ω and δ ai (A j ) is the Dirac measure, where⎧⎪⎨ 1 if a i ∈ A jδ ai (A j ) =.⎪⎩ 0 if a i /∈ A j( n∑)n∑Conditional on (F (A 1 ), · · · , F (A K )), the vector δ ai (A 1 ), · · · , δ ai (A K ) has a multinominaldistribution. By the conjugacy of Dirichlet distribution to the multi-nominali=1i=1distribution,the posterior distribution of (F (A 1 ), · · · , F (A K )) is still a Dirichlet distribution(F (A 1 ), · · · , F (A K )) | a ∼ Dir(α 0 G 0 (A 1 ) +n∑δ ai (A 1 ), · · · , α 0 G 0 (A K ) +i=1)n∑δ ai (A K )Because this result is valid for any finite measurable partition, the posterior of F is still Dirichletprocess by definition, with new parameters α ∗ 0 and G∗ 0 , wherei=1α ∗ 0 = α 0 + nG ∗ 0 = α 0α 0 + n G 0 +nα 0 + nThe posterior shape parameter, G ∗ 0 , is the mixture of the prior and the empirical distributionimplied by observations. As n → ∞, the shape parameter of the posterior converges tothe empirical distribution. The concentration parameter α ∗ 0n∑i=1δ ain→ ∞ implies the posterior of Fconverges to the empirical distribution with probability one. Ferguson (1973) showed that arandom distribution drawn from a Dirichlet process is almost sure discrete, although the shapeparameter G 0 can be continuous.27