  Gaussian Bayesian networks (a.k.a. linear Gaussian structural equation
models) are widely used to model causal interactions among continuous
variables. In this work, we study the problem of learning a fixed-structure
Gaussian Bayesian network up to a bounded error in total variation distance. We
analyze the commonly used node-wise least squares regression (LeastSquares) and
prove that it has a near-optimal sample complexity. We also study a couple of
new algorithms for the problem:
  - BatchAvgLeastSquares takes the average of several batches of least squares
solutions at each node, so that one can interpolate between the batch size and
the number of batches. We show that BatchAvgLeastSquares also has near-optimal
sample complexity.
  - CauchyEst takes the median of solutions to several batches of linear
systems at each node. We show that the algorithm specialized to polytrees,
CauchyEstTree, has near-optimal sample complexity.
  Experimentally, we show that for uncontaminated, realizable data, the
LeastSquares algorithm performs best, but in the presence of contamination or
DAG misspecification, CauchyEst/CauchyEstTree and BatchAvgLeastSquares
respectively perform better.
