# Notes on: Gorham, J., & Mackey, L. (2015): Measuring Sample Quality With Stein's Method

## Overview

1. Motivates Steins method
2. Steins method
• , the function selected in the maximization process of the Stein Discrepancy gives us a method, can be intuitively thought of as being a feature selection function which finds the features producing the most discrepancy between our approx. density and the actual density
3. Classical Stein Set and Discrepancy
• Methods for constructing a Stein operator for sufficiently smooth functions
• Proves upper bound for classical Stein Discrepancy
4. Methods for computing Stein Discrepancies
• Graph Stein set
• Spanner Stein discrepancies using spanner graphs
• Linear programs used to solve the finite-dimensional subproblems of maximizing the function in each of the components to obtain the Stein discrepancy
5. Experiments

## Notation

• Often refer to a generic norm on with associated dual norms for vectors
• is target distribution with open convex support
• continuously differentiable density
• probability mass function induces a discrete distribution and approx.

for any target expHectation .

• weighted sample of distinct sample points with weights encoded in the probability mass function
• real-valued operator
• set of valued functions

## Stuff

Construct quality of measure with following properties:

1. detects when a sequence of samples is converging to the target
2. detects when a sequence of sample is not converging to the target
3. computationally feasible

First consider expected deviation between sample and target expectations over a class of real-valued test functions :

• If class of test functions is sufficiently large => implies that the seuqence of sample measures converges weakly to

Varying class of test functions of IPM we recover many well-known probability metrics:

## Stein's method

1. Identify a real-valued operator acting on a set of valued functions of for which

Together, and define the Stein discrepancy,

an IPM quality measure with no explicit integration under .

2. Lower bound the Stein discrepancy by a familiar convergence-determining IPM
• Can be perforemd once, in advance, for alrge classes of target distributions and ensures that, for any sequence of probability measures , converges to zero if and only if
3. Upper bound the Stein discrepancy by any means necessary to demonstrate convergence to zero under suitable conditions.

### Challenges

• Constructing a Stein operator which produce mean-zero functions under

### Identifying a Stein operator

If we let:

• denote the boundary of (an empty set when
• represent the outward unit normal vector to the boundary at

then we may define the classical Stein set

of sufficiently smooth functions satisfying a Neumann-type boundary condition (referring to the inner product of the function and the outward unit vector ).

From this they get the following proposition

If , then for all .

Together, and form the classical Stein discrepancy , which is the main study of the paper.