Don Fraser has given an interesting account of the agreements and
disagreements between Bayesian posterior probabilities and confidence levels.
In this comment I discuss some cases where the lack of such agreement is
extreme. I then discuss a few cases where it is possible to have Bayes
procedures with frequentist validity. Such frequentist-Bayesian---or
Frasian---methods deserve more attention [arXiv:1112.5582].

We propose a relaxed privacy definition called {\em random differential
privacy} (RDP). Differential privacy requires that adding any new observation
to a database will have small effect on the output of the data-release
procedure. Random differential privacy requires that adding a {\em randomly
drawn new observation} to a database will have small effect on the output. We
show an analog of the composition property of differentially private procedures
which applies to our new definition.

We investigate and extend the conformal prediction method due to
Vovk,Gammerman and Shafer (2005) to construct nonparametric prediction regions.
These regions have guaranteed distribution free, finite sample coverage,
without any assumptions on the distribution or the bandwidth. Explicit
convergence rates of the loss function are established for such regions under
standard regularity conditions. Approximations for simplifying implementation
and data driven bandwidth selection methods are also discussed. The theoretical
properties of our method are demonstrated through simulations.

High density clusters can be characterized by the connected components of a
level set $L(\lambda) = \{x:\ p(x)>\lambda\}$ of the underlying probability
density function $p$ generating the data, at some appropriate level
$\lambda\geq 0$. The complete hierarchical clustering can be characterized by a
cluster tree ${\cal T}= \bigcup_{\lambda} L(\lambda)$. In this paper, we study
the behavior of a density level set estimate $\widehat L(\lambda)$ and cluster
tree estimate $\widehat{\cal{T}}$ based on a kernel density estimator with
kernel bandwidth $h$.

We sharply characterize the performance of different penalization schemes for
the problem of selecting the relevant variables in the multi-task setting.
Previous work focuses on the regression problem where conditions on the design
matrix complicate the analysis. A clearer and simpler picture emerges by
studying the Normal means model. This model, often used in the field of
statistics, is a simplified model that provides a laboratory for studying
complex procedures.

Undirected graphical models encode in a graph $G$ the dependency structure of
a random vector $Y$. In many applications, it is of interest to model $Y$ given
another random vector $X$ as input. We refer to the problem of estimating the
graph $G(x)$ of $Y$ conditioned on $X=x$ as ``graph-valued regression.'' In
this paper, we propose a semiparametric method for estimating $G(x)$ that
builds a tree on the $X$ space just as in CART (classification and regression
trees), but at each leaf of the tree estimates a graph. We call the method
``Graph-optimized CART,'' or Go-CART.

We develop nonparametric methods for estimating filamentary structure from
planar point process data and find the minimax lower bound for this problem. We
show that, under weak conditions, the filaments have a simple geometric
representation as the medial axis of the data distribution's support. Our
methods convert an estimator of the support's boundary into an estimator of the
filaments. We find the rates of convergence of our estimators and show that
when using an optimal boundary estimator, they achieve the minimax rate.

We study graph estimation and density estimation in high dimensions. To avoid
the curse of dimensionality, we consider a family of density estimators based
on tree structured undirected graphical models. We do not assume the true
distribution corresponds to a tree; rather, we try to find the best tree-based
approximation to the true distribution. We apply the Chow-Liu algorithm to
kernel density estimates to build a tree and then use a data-splitting scheme
to choose the number of edges. We also prove oracle properties on both function
estimation and structure learning.

The lasso has become an important practical tool for high dimensional
regression as well as the object of intense theoretical investigation. But
despite the availability of efficient algorithms, the lasso remains
computationally demanding in regression problems where the number of variables
vastly exceeds the number of data points. A much older method, marginal
regression, largely displaced by the lasso, offers a promising alternative in
this case. Computation for marginal regression is practical even when the
dimension is very high.

We consider the problem of reliably finding filaments in point clouds.
Realistic data sets often have numerous filaments of various sizes and shapes.
Statistical techniques exist for finding one (or a few) filaments but these
methods do not handle noisy data sets with many filaments. Other methods can be
found in the astronomy literature but they do not have rigorous statistical
guarantees. We propose the following method. Starting at each data point we
construct the steepest ascent path along a kernel density estimator.