A recursive procedure for density estimation on the binary hypercube

Maxim Raginsky, Jorge G. Silva, Svetlana Lazebnik, Rebecca Willett

Research output: Contribution to journalArticlepeer-review


This paper describes a recursive estimation procedure for multivariate binary densities (probability distributions of vectors of Bernoulli random variables) using orthogonal expansions. For d covariates, there are 2d basis coefficients to estimate, which renders conventional approaches computationally prohibitive when d is large. However, for a wide class of densities that satisfy a certain sparsity condition, our estimator runs in probabilistic polynomial time and adapts to the unknown sparsity of the underlying density in two key ways: (1) it attains near-minimax mean-squared error for moderate sample sizes, and (2) the computational complexity is lower for sparser densities. Our method also allows for flexible control of the trade-off between mean-squared error and computational complexity.

Original languageEnglish (US)
Pages (from-to)820-858
Number of pages39
JournalElectronic Journal of Statistics
Issue number1
StatePublished - 2013


  • Adaptive estimation
  • Binary hypercube
  • Density estimation
  • Minimax estimation
  • Sparsity
  • Walsh basis

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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