We obtain the maximum likelihood estimator of the central subspace under conditional normality of the predictors given the response. Analytically and in simulations we found that our new estimator can preform much better than sliced inverse regression, sliced average variance estimation and directional regression, and that it seems quite robust to deviations from normality.

Partial dimension reduction is a general method to seek informative convex combinations of predictors of primary interest, which includes dimension reduction as its special case when the predictors in the remaining part are constants. In this paper, we propose a novel method to conduct partial dimension reduction estimation for predictors of primary interest without assuming that the remaining ...

In the previous lecture we defined Sufficient Statistics, which capture all information in the data relevant to estimating a desired parameter (hopefully, in a concise way). Motivated by the fact that useful sufficient statistics (i.e., those with dimension independent of sample size) only exist for certain (i.e., exponential family) distributions, we defined two more general notions, the Rate ...

In this paper, we propose a post randomization technique to learn a Bayesian network (BN) from distributed heterogeneous data, in a privacy sensitive fashion. In this case, two or more parties own sensitive data but want to learn a Bayesian network from the combined data. We consider both structure and parameter learning for the BN. The only required information from the data set is a set of su...

We investigate the problem of learning discrete, undirected graphical models in a differentially private way. We show that the approach of releasing noisy sufficient statistics using the Laplace mechanism achieves a good trade-off between privacy, utility, and practicality. A naive learning algorithm that uses the noisy sufficient statistics “as is” outperforms general-purpose differentially pr...

This paper studies two-player zero-sum repeated Bayesian games in which every player has a private type that is unknown to the other player, and the initial probability of the type of every player is publicly known. The types of players are independently chosen according to the initial probabilities, and are kept the same all through the game. At every stage, players simultaneously choose actio...

Optimizing executions of queries is the ultimate goal of the query optimizer. Unfortunately, due to the complexities of queries, accuracy of statistics, validities of assumptions, etc., query optimizers often cannot find the best execution plans in their search spaces, conveniently called the optimal plans, for the queries. In this paper, we develop a comprehensive framework for re-optimization...

Assume submodels are in exponential family: pm(x, z; θm) = exp { θ m ( φm(x)φ Z(z) ) −Am(θm) } for x ∈ X , z ∈ Zm and 0 otherwise Reformulation of Product EM: Aggregate parameters: b = ∑ m bm, bm = φ X m(x) θm E-step: compute expected sufficient statistics μ = E(b,∩mZm) def = Eq(z;b)φ(z) with support ∩mZm M-step: set θm to match moments φ X m(x)μ Exponential family formulation Two sources of in...

Necessary and sufficient conditions are given for a triangular array of numbers to be expectations of order statistics of some nonnegative random variable. Using well-known recurrence relations, the expectations of all order statistics of the largest sample size, n, in the triangular array, or the expectations of the smallest of every sample size up to and including n are sufficient to determin...