Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing

Constantinos Daskalakis, Qinxuan Pan
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:697-703, 2017.

Abstract

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, $H^2(P,Q)$, between $P$ and $Q$ is upper bounded by the sum, $\sum_v H^2(P_{v} ∪\Pi_v, Q_{v} ∪\Pi_v)$, of the square Hellinger distances between the marginals of $P$ and $Q$ on every node $v$ and its parents $\Pi_v$ 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 (unknown) Bayesian networks $P$ and $Q$ on the same (but potentially unknown) DAG satisfy $P=Q$ vs $d_\rm TV(P,Q)>ε$ can be performed from $\tilde{O}(|Σ|^3/4(d+1) ⋅n/ε^2)$ 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 $\tilde{O}(|Σ|^4.5 n/ε^2)$. In both cases the dependence of the sample complexity 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 $O(\sqrt{n}/ε^2)$, which is optimal up to constant factors.

Cite this Paper

BibTeX
@InProceedings{pmlr-v65-daskalakis17a, title = {Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing}, author = {Daskalakis, Constantinos and Pan, Qinxuan}, booktitle = {Proceedings of the 2017 Conference on Learning Theory}, pages = {697--703}, year = {2017}, editor = {Kale, Satyen and Shamir, Ohad}, volume = {65}, series = {Proceedings of Machine Learning Research}, month = {07--10 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v65/daskalakis17a/daskalakis17a.pdf}, url = {https://proceedings.mlr.press/v65/daskalakis17a.html}, abstract = {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, $H^2(P,Q)$, between $P$ and $Q$ is upper bounded by the sum, $\sum_v H^2(P_{v} ∪\Pi_v, Q_{v} ∪\Pi_v)$, of the square Hellinger distances between the marginals of $P$ and $Q$ on every node $v$ and its parents $\Pi_v$ 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 (unknown) Bayesian networks $P$ and $Q$ on the same (but potentially unknown) DAG satisfy $P=Q$ vs $d_\rm TV(P,Q)>ε$ can be performed from $\tilde{O}(|Σ|^3/4(d+1) ⋅n/ε^2)$ 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 $\tilde{O}(|Σ|^4.5 n/ε^2)$. In both cases the dependence of the sample complexity 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 $O(\sqrt{n}/ε^2)$, which is optimal up to constant factors.} }
Endnote
%0 Conference Paper %T Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing %A Constantinos Daskalakis %A Qinxuan Pan %B Proceedings of the 2017 Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2017 %E Satyen Kale %E Ohad Shamir %F pmlr-v65-daskalakis17a %I PMLR %P 697--703 %U https://proceedings.mlr.press/v65/daskalakis17a.html %V 65 %X 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, $H^2(P,Q)$, between $P$ and $Q$ is upper bounded by the sum, $\sum_v H^2(P_{v} ∪\Pi_v, Q_{v} ∪\Pi_v)$, of the square Hellinger distances between the marginals of $P$ and $Q$ on every node $v$ and its parents $\Pi_v$ 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 (unknown) Bayesian networks $P$ and $Q$ on the same (but potentially unknown) DAG satisfy $P=Q$ vs $d_\rm TV(P,Q)>ε$ can be performed from $\tilde{O}(|Σ|^3/4(d+1) ⋅n/ε^2)$ 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 $\tilde{O}(|Σ|^4.5 n/ε^2)$. In both cases the dependence of the sample complexity 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 $O(\sqrt{n}/ε^2)$, which is optimal up to constant factors.
APA
Daskalakis, C. & Pan, Q.. (2017). Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing. Proceedings of the 2017 Conference on Learning Theory, in Proceedings of Machine Learning Research 65:697-703 Available from https://proceedings.mlr.press/v65/daskalakis17a.html.

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