Archive for Chinese restaurant process

Xi’an street food [jatp]

Posted in pictures, Travel with tags , , , , , , on October 27, 2019 by xi'an

Sens’o risotto

Posted in Kids, pictures, Wines with tags , , , , , , on February 6, 2016 by xi'an

A chance dinner last Sunday led us to dine in Sens’o, an empty Italian restaurant on one of the Paris islands with a superlative scallops risotto and a chance encounter resulted from talking from the idle waitress who happened to be a free-lance journalist. We talked for quite a while on her previous trips to Haïti, Egypt, and field hospitals at the Syrian border in Turkey. (The restaurant was empty for a combination of reasons, from the drop in tourists after the November 13 killings to January being a low tide month, to a blistery Sunday night being unattractive for revellers. Not because of bad reviews…)

mixture models with a prior on the number of components

Posted in Books, Statistics, University life with tags , , , , , , , on March 6, 2015 by xi'an

mixdir

“From a Bayesian perspective, perhaps the most natural approach is to treat the numberof components like any other unknown parameter and put a prior on it.”

Another mixture paper on arXiv! Indeed, Jeffrey Miller and Matthew Harrison recently arXived a paper on estimating the number of components in a mixture model, comparing the parametric with the non-parametric Dirichlet prior approaches. Since priors can be chosen towards agreement between those. This is an obviously interesting issue, as they are often opposed in modelling debates. The above graph shows a crystal clear agreement between finite component mixture modelling and Dirichlet process modelling. The same happens for classification.  However, Dirichlet process priors do not return an estimate of the number of components, which may be considered a drawback if one considers this is an identifiable quantity in a mixture model… But the paper stresses that the number of estimated clusters under the Dirichlet process modelling tends to be larger than the number of components in the finite case. Hence that the Dirichlet process mixture modelling is not consistent in that respect, producing parasite extra clusters…

In the parametric modelling, the authors assume the same scale is used in all Dirichlet priors, that is, for all values of k, the number of components. Which means an incoherence when marginalising from k to (k-p) components. Mild incoherence, in fact, as the parameters of the different models do not have to share the same priors. And, as shown by Proposition 3.3 in the paper, this does not prevent coherence in the marginal distribution of the latent variables. The authors also draw a comparison between the distribution of the partition in the finite mixture case and the Chinese restaurant process associated with the partition in the infinite case. A further analogy is that the finite case allows for a stick breaking representation. A noteworthy difference between both modellings is about the size of the partitions

\mathbb{P}(s_1,\ldots,s_k)\propto\prod_{j=1}^k s_j^{-\gamma}\quad\text{versus}\quad\mathbb{P}(s_1,\ldots,s_k)\propto\prod_{j=1}^k s_j^{-1}

in the finite (homogeneous partitions) and infinite (extreme partitions) cases.

An interesting entry into the connections between “regular” mixture modelling and Dirichlet mixture models. Maybe not ultimately surprising given the past studies by Peter Green and Sylvia Richardson of both approaches (1997 in Series B and 2001 in JASA).