De Blasi 2015 - Gibbs-Type Priors - Overview

Summary

De Blasi, Favaro, Lijoi, Mena and Prunster review Gibbs-type priors: discrete nonparametric priors whose probability of a new value depends on the sample size $n$ and the number of distinct values $k$ but not on the frequencies. They contain the Dirichlet process (DP), Pitman-Yor (PY) and normalised generalised gamma (NGG) processes, with a discount $\sigma$ controlling whether $K_n$ grows like $\log n$, $n^{\sigma }$ or saturates. Key takeaway: a prior on the number of clusters flatter than the DP’s is available in closed form, and the same EPPF gives exact unseen-species predictions.

Overview

Research question. Beyond the Dirichlet process, what is a tractable and principled family of discrete random probability measures, and what do its exchangeable partitions say about clustering, prediction of new species and consistency?

Contribution. (i) A predictive classification of species sampling models (Prop. 1); (ii) the product-form EPPF $V_{n,k}\prod (1-\sigma {)}_{n_i-1}$; (iii) special cases (DP, PY, NGG, Gnedin); (iv) effects on mixture-model clustering; (v) Bayesian nonparametric estimators for species variety and discovery probabilities; (vi) frequentist consistency and dependent extensions.

Key results. $P(\text{new}\mid \cdot )=V_{n+1,k+1}/V_{n,k}$; PY: $(\theta +\sigma k)/(\theta +n)$; $K_n/c_n(\sigma )\to S_{\sigma }$ a.s.; PY estimator ${\hat{K}}_m^{(n)}=(k+\theta /\sigma )[\frac{(\theta +n+\sigma {)}_m}{(\theta +n{)}_m}-1]$; consistency at discrete $P_0$ but inconsistency at diffuse $P_0$ unless $\sigma =0$.

Source Map

Paper section	Covered in
Abstract; 1 Introduction	this note
1.1 Discrete measures, EPPF, predictive distributions (Eqs. 1-11)	De Blasi 2015 - EPPF and Predictive Structure
1.2 Applications to species sampling and mixtures	De Blasi 2015 - EPPF and Predictive Structure
1.3 Outline	this note
2 Gibbs-type priors (Prop. 1, Eqs. 13-17, Remark 2)	De Blasi 2015 - Gibbs-Type Priors Definition and Classification
2.1 Product partition models	De Blasi 2015 - Gibbs-Type Priors Definition and Classification
2.2 Sub-classes (PY, sigma-stable, finite species, Gnedin, NGG, support)	De Blasi 2015 - Subclasses Pitman-Yor NGG and Finite-Species Priors
3 Hierarchical mixtures (Eqs. 26-27, Figs 1-4, Table 1)	De Blasi 2015 - Mixture Models and the Prior on the Number of Clusters
4 Prediction in species sampling (4.1 overall, 4.2 rare; Eqs. 28-43, Table 2, Fig. 5)	De Blasi 2015 - Species Sampling and Unseen Species Prediction
5 Frequentist asymptotics (5.1 general, 5.2 illustrations)	De Blasi 2015 - Consistency, Dependent Processes and Conclusions
6 Dependent processes; 7 Concluding remarks	De Blasi 2015 - Consistency, Dependent Processes and Conclusions
Appendix (proof of Prop. 1); References [1]-[76]	De Blasi 2015 - Gibbs-Type Priors Definition and Classification

Motivating PyMC examples

• E30 Exchangeable partitions: Polya urns, the Chinese restaurant process, Pitman-Yor and unseen species (Moby-Dick word tokens; Barro Colorado Island 50-ha tree census). The listed method string includes: urn simulation and de Finetti limits, the EPPF checked by brute force, NUTS on $(\alpha ,d)$ with no latent assignments, stable log rising factorial, frequency-of-frequencies and Heaps-curve PPCs, exact held-out predictive from EPPF ratios.
• E26 (DP mixtures; prior on $K$ and sensitivity to $\alpha$); E75 (image segmentation with unknown regions). URL: PyMC challenges examples.

Relevance to marketing measurement

Unseen-species estimation transfers to “how many new items/customers/creatives will appear”: e.g. expected number of new SKUs bought in a future sample, or share of future transactions that are first-time item purchases. PY priors also give heavy-tailed cluster sizes, matching the long tail of product popularity. Use with caution: the predictions assume exchangeability, which fails with trends and seasonality.

Connections

Teh 2010 - Dirichlet Process - Overview, Teh 2010 - Extensions and Reference Map, Nonparametric Models Overview, Pareto-NBD Model (customer-base “unseen” counts, different mechanism).

See Also

De Blasi 2015 index