  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 $2^d$ 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.
