Standard Sampling Models Research Articles

Property Testing of Joint Distributions using Conditional Samples

In this article, we consider the problem of testing properties of joint distributions under the Conditional Sampling framework. In the standard sampling model, sample complexity of testing properties of joint distributions are exponential in the dimension, resulting in inefficient algorithms for practical use. While recent results achieve efficient algorithms for product distributions with significantly smaller sample complexity, no efficient algorithm is expected when the marginals are not independent. In this article, we initialize the study of conditional sampling in the multidimensional setting. We propose a subcube conditional sampling model where the tester can condition on a (adaptively) chosen subcube of the domain. Due to its simplicity, this model is potentially implementable in many practical applications, particularly when the distribution is a joint distribution over Σ n for some set Σ. We present algorithms for various fundamental properties of distributions in the subcube-conditioning model and prove that the sample complexity is polynomial in the dimension n (and not exponential as in the traditional model). We present an algorithm for testing identity to a known distribution using Õ( n 2 )-subcube-conditional samples, an algorithm for testing identity between two unknown distributions using Õ( n 5 )-subcube-conditional samples and an algorithm for testing identity to a product distribution using Õ( n 5 )-subcube-conditional samples. The central concept of our technique involves an elegant chain rule, which can be proved using basic techniques of probability theory, yet it is powerful enough to avoid the curse of dimensionality.

A Chasm Between Identity and Equivalence Testing with Conditional Queries

A recent model for property testing of probability distributions [CFGM13, CRS15] enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain. In particular, Canonne, Ron, and Servedio [CRS15] showed that, in this setting, testing identity of an unknown distribution D (i.e., whether D = D∗ for an explicitly known D∗) can be done with a constant number of samples, independent of the support size n – in contrast to the required √ n in the standard sampling model. However, it was unclear whether the same held for the case of testing equivalence, where both distributions are unknown. Indeed, while Canonne, Ron, and Servedio [CRS15] established a polylog(n)-query upper bound for equivalence testing, very recently brought down to O(log logn) by Falahatgar et al. [FJO+15], whether a dependence on the domain size n is necessary was still open, and explicitly posed by Fischer at the Bertinoro Workshop on Sublinear Algorithms [Sublinear.info, Problem 66]. In this work, we answer the question in the positive, showing that any testing algorithm for equivalence must make Ω (√ log logn ) queries in the conditional sampling model. Interestingly, this demonstrates an intrinsic qualitative gap between identity and equivalence testing, absent in the standard sampling model (where both problems have sampling complexity nΘ(1)). Turning to another question, we investigate the complexity of support size estimation. We provide a doubly-logarithmic upper bound for the adaptive version of this problem, generalizing work of Ron and Tsur [RT14] to our weaker model. We also establish a logarithmic lower bound for the non-adaptive version of this problem. This latter result carries on to the related problem of non-adaptive uniformity testing, an exponential improvement over previous results that resolves an open question of Chakraborty, Fischer, Goldhirsh, and Matsliah [CFGM13]. ∗EECS, MIT. Email: jayadev@csail.mit.edu. Research supported by grant from MITEI-Shell program. †Columbia University. Email: ccanonne@cs.columbia.edu. Research supported by NSF CCF-1115703 and NSF CCF-1319788. ‡EECS, MIT. Email: g@csail.mit.edu.

Optimal prediction of the number of unseen species

Estimating the number of unseen species is an important problem in many scientific endeavors. Its most popular formulation, introduced by Fisher et al. [Fisher RA, Corbet AS, Williams CB (1943) J Animal Ecol 12(1):42-58], uses n samples to predict the number U of hitherto unseen species that would be observed if [Formula: see text] new samples were collected. Of considerable interest is the largest ratio t between the number of new and existing samples for which U can be accurately predicted. In seminal works, Good and Toulmin [Good I, Toulmin G (1956) Biometrika 43(102):45-63] constructed an intriguing estimator that predicts U for all [Formula: see text] Subsequently, Efron and Thisted [Efron B, Thisted R (1976) Biometrika 63(3):435-447] proposed a modification that empirically predicts U even for some [Formula: see text], but without provable guarantees. We derive a class of estimators that provably predict U all of the way up to [Formula: see text] We also show that this range is the best possible and that the estimator's mean-square error is near optimal for any t Our approach yields a provable guarantee for the Efron-Thisted estimator and, in addition, a variant with stronger theoretical and experimental performance than existing methodologies on a variety of synthetic and real datasets. The estimators are simple, linear, computationally efficient, and scalable to massive datasets. Their performance guarantees hold uniformly for all distributions, and apply to all four standard sampling models commonly used across various scientific disciplines: multinomial, Poisson, hypergeometric, and Bernoulli product.