Publications

2023
Article

Which Models Have Perceptually-Aligned Gradients? An Explanation via Off-Manifold Robustness

By: Suraj Srinivas, Sebastian Bordt and Himabindu Lakkaraju

One of the remarkable properties of robust computer vision models is that their input-gradients are often aligned with human perception, referred to in the literature as perceptually-aligned gradients (PAGs). Despite only being trained for classification, PAGs cause... View Details

Keywords: AI and Machine Learning; Mathematical Methods

March 2020
Article

Diagnosing Missing Always at Random in Multivariate Data

By: Iavor I. Bojinov, Natesh S. Pillai and Donald B. Rubin

Models for analyzing multivariate data sets with missing values require strong, often assessable, assumptions. The most common of these is that the mechanism that created the missing data is ignorable—a twofold assumption dependent on the mode of inference. The first... View Details

Keywords: Missing Data; Diagnostic Tools; Sensitivity Analysis; Hypothesis Testing; Missing At Random; Row Exchangeability; Analytics and Data Science; Mathematical Methods

June 2000
Article

On the Regulatory Application of Efficiency Measures

The last decade has witnessed a change to more powerful incentive schemes and the adoption by a large number of regulators of some form of price cap regimes. The efficiency frontiers literature tackles the problem of measuring the X factor in a price cap regime... View Details

Keywords: Performance Productivity; Mathematical Methods; Performance Efficiency

August 2001 (Revised July 2008)
Technical Note

A Technical Note and Discussion on Real Estate Valuation (IBET): Back of the Envelope (BOE) on Bonhomme Place: A Case within a Case

By: Arthur I Segel

Discusses real estate valuation. Reviews "back of the envelope" valuation; real estate appraisal methods, including the income method; market comparables and replacement costs; and more complex computer modeling. Also discusses other variables that could influence... View Details

Keywords: Investment Funds; Mathematical Methods; Valuation; Real Estate Industry

Article

The Shapley Value as a von Neumann-Morgenstern Utility

By: A. E. Roth

Keywords: Value; Mathematical Methods

Article

Portfolio Value-at-Risk Optimization for Asymmetrically Distributed Asset Returns

By: Joel Goh, Kian Guan Lim, Melvyn Sim and Weina Zhang

We propose a new approach to portfolio optimization by separating asset return distributions into positive and negative half-spaces. The approach minimizes a newly-defined Partitioned Value-at-Risk (PVaR) risk measure by using half-space statistical information. Using... View Details

Keywords: Robust Optimization; Portfolio Management; Value-at-risk; Mathematical Methods; Finance

July 1985 (Revised March 1994)
Background Note

Exposure and Hedging

By: David E. Bell

Describes the concept of exposure; the dependence of a goal on an uncertain external event. Describes in detail how hedges may be constructed to eliminate exposure, including the algebra of cross-hedging and hedge ratios. The relevance of regression analysis is... View Details

Keywords: Mathematical Methods; Finance

1976
Book

Lies, Damn Lies and Statistics: The Manipulation of Public Opinion in America

By: Michael A. Wheeler

Keywords: Society; Mathematical Methods; United States

01 Dec 2007
News

From A to Z

French Last Job: Business Analyst, McKinsey & Company, Moscow Age: 26 Why HBS? “My degree is in economics, which was exciting but strictly mathematical and theoretical. I wanted to become a more well-rounded individual and gain a wider... View Details

Keywords: Business Schools & Computer & Management Training; Educational Services

Web

Prelude - Option Pricing in Theory & Practice: The Nobel Prize Research of Robert C. Merton - Exhibits - Historical Collections

economics through his working relationship with Paul Samuelson. Perhaps more than any other economist to date, Paul Samuelson is credited with ushering in the modern era of mathematical economic analysis incorporating formal probability... View Details

July–August 2023
Article

Demand Learning and Pricing for Varying Assortments

By: Kris Ferreira and Emily Mower

Problem Definition: We consider the problem of demand learning and pricing for retailers who offer assortments of substitutable products that change frequently, e.g., due to limited inventory, perishable or time-sensitive products, or the retailer’s desire to... View Details

Keywords: Experiments; Pricing And Revenue Management; Retailing; Demand Estimation; Pricing Algorithm; Marketing; Price; Demand and Consumers; Mathematical Methods

July 1990
Background Note

Perceptual Mapping: A Manager's Guide

By: Robert J. Dolan

Describes the perceptual mapping technique in a non-technical fashion. The procedure is useful for the depiction of the structure of the market. Discusses alternative methods, presents examples of each, and shows how the maps can be used in marketing decision making. View Details

Keywords: Mathematical Methods; Industry Structures; Marketing

1999
Other Unpublished Work

Estimating Industry Multiples

By: Malcolm Baker and R. S. Ruback

We analyze industry multiples for the S&P 500 in 1995. We use Gibbs sampling to estimate simultaneously the error specification and small sample minimum variance multiples for 22 industries. In addition, we consider the performance of four common multiples: the simple... View Details

Keywords: Management Analysis, Tools, and Techniques; Performance; Mathematical Methods

July 2023
Article

Design and Analysis of Switchback Experiments

By: Iavor I Bojinov, David Simchi-Levi and Jinglong Zhao

In switchback experiments, a firm sequentially exposes an experimental unit to a random treatment, measures its response, and repeats the procedure for several periods to determine which treatment leads to the best outcome. Although practitioners have widely adopted... View Details

Keywords: Switchback Experiments; Design; Analysis; Mathematical Methods

January 1997
Background Note

Simulation as a Decision Aid

By: Roy D. Shapiro

A brief introduction to simulation--what it is, why it's used, etc. Meant to set context for a first class on simulation. A rewritten version of an earlier note. View Details

Keywords: Mathematical Methods; Decision Making