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We show that the square Hellinger distance between two Bayesian networks on the same directed graph, G, is subadditive with respect to the neighborhoods of G. Namely, if P and Q are the probability distributions defined by two Bayesian networks on the same DAG, our inequality states that the square Hellinger distance, H2(P,Q), between P and Q is upper bounded by the sum, ∑vH2(P{v}∪Πv,Q{v}∪Πv), of the square Hellinger distances between the marginals of P and Q on every node v and its parents Πv in the DAG. Importantly, our bound does not involve the conditionals but the marginals of P and Q. We derive a similar inequality for more general Markov Random Fields. As an application of our inequality, we show that distinguishing whether two Bayesian networks P and Q on the same (but potentially unknown) DAG satisfy P=Q vs dTV(P,Q)>ϵ can be performed from Õ (|Σ|3/4(d+1)⋅n/ϵ2) samples, where d is the maximum in-degree of the DAG and Σ the domain of each variable of the Bayesian networks. If P and Q are defined on potentially different and potentially unknown trees, the sample complexity becomes Õ (|Σ|4.5n/ϵ2), whose dependence on n,ϵ is optimal up to logarithmic factors. Lastly, if P and Q are product distributions over {0,1}n and Q is known, the sample complexity becomes O(n‾√/ϵ2), which is optimal up to constant factors.
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Optimal Stopping Rules for Sequential Hypothesis Testing
Suppose that we are given sample access to an unknown distribution p over n elements and an explicit distribution q over the same n elements. We would like to reject the null hypothesis“p=q” after seeing as few samples as possible, whenp6=q, while we never want to reject the null, when p=q. Well-known results show thatΘ(√n/2)samples are necessary and sufficient for distinguishing whether p equals q versus p is-far from q in total variation distance. However,this requires the distinguishing radiusto be fixed prior to deciding how many samples to request.Our goal is instead to design sequential hypothesis testers, i.e. online algorithms that request i.i.d.samples from p and stop as soon as they can confidently reject the hypothesis p=q, without being given a lower bound on the distance between p and q, whenp6=q. In particular, we want to minimize the number of samples requested by our tests as a function of the distance between p and q, and if p=q we want the algorithm, with high probability, to never reject the null. Our work is motivated by and addresses the practical challenge of sequential A/B testing in Statistics.We show that, when n= 2, any sequential hypothesis test must seeΩ(1dtv(p,q)2log log1dtv(p,q))samples, with high (constant) probability, before it rejects p=q, where dtv(p,q) is the—unknown to the tester—total variation distance between p and q. We match the dependence of this lower bound ondtv(p,q)by proposing a sequential tester that rejects p=q from at most O(√ndtv(p,q)2log log1dtv(p,q))samples with high (constant) probability. TheΩ(√n)dependence on the support size n is also known to be necessary. We similarly provide two-sample sequential hypothesis testers, when sample access is given to both p and q, and discuss applications to sequential A/B testing.
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- Award ID(s):
- 1650733
- PAR ID:
- 10086307
- Date Published:
- Journal Name:
- 25th Annual European Symposium on Algorithms (ESA)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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