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  • All HBS Web  (1,959)
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    • News  (215)
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  • Article

Active World Model Learning with Progress Curiosity

By: Kuno Kim, Megumi Sano, Julian De Freitas, Nick Haber and Daniel Yamins
World models are self-supervised predictive models of how the world evolves. Humans learn world models by curiously exploring their environment, in the process acquiring compact abstractions of high bandwidth sensory inputs, the ability to plan across long temporal... View Details
Keywords: World Models; Mathematical Methods
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Kim, Kuno, Megumi Sano, Julian De Freitas, Nick Haber, and Daniel Yamins. "Active World Model Learning with Progress Curiosity." Proceedings of the International Conference on Machine Learning (ICML) 37th (2020).
  • Article

Matching in Networks with Bilateral Contracts: Corrigendum

By: John William Hatfield, Ravi Jagadeesan and Scott Duke Kominers
Hatfield and Kominers (2012) introduced a model of matching in networks with bilateral contracts and showed that stable outcomes exist in supply chains when firms' preferences over contracts are fully substitutable. Hatfield and Kominers (2012) also asserted that in... View Details
Keywords: Matching With Contracts; Substitutability; Mathematical Methods
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Hatfield, John William, Ravi Jagadeesan, and Scott Duke Kominers. "Matching in Networks with Bilateral Contracts: Corrigendum." American Economic Journal: Microeconomics 12, no. 3 (August 2020): 277–285.
  • 2015
  • Working Paper

Configurations of Extremal Type II Codes

By: Noam D. Elkies and Scott Duke Kominers
We prove configuration results for extremal Type II codes, analogous to the configuration results of Ozeki and of the second author for extremal Type II lattices. Specifically, we show that for n∈{8,24,32,48,56,72,96} every extremal Type II code of length n is... View Details
Keywords: Mathematical Methods
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Elkies, Noam D., and Scott Duke Kominers. "Configurations of Extremal Type II Codes." Working Paper, March 2015.
  • February 1991 (Revised February 1993)
  • Background Note

Regression Analysis

By: David E. Bell
Provides a relatively simple introduction to multivariate regression analysis. View Details
Keywords: Mathematical Methods
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Bell, David E. "Regression Analysis." Harvard Business School Background Note 191-117, February 1991. (Revised February 1993.)
  • Article

Improved Bounds on the Sizes of S.P Numbers

By: Paul Myer Kominers and Scott Duke Kominers
A number which is S.P in base r is a positive integer which is equal to the sum of its base-r digits multiplied by the product of its base-r digits. These numbers have been studied extensively in The Mathematical Gazette. Recently, Shah Ali... View Details
Keywords: Mathematical Methods
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Kominers, Paul Myer, and Scott Duke Kominers. "Improved Bounds on the Sizes of S.P Numbers." Mathematical Gazette 94, no. 529 (March 2010): 127–129.
  • Article

Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)

By: Scott Duke Kominers and Zachary Abel
We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1. View Details
Keywords: Mathematical Methods
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Kominers, Scott Duke, and Zachary Abel. "Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)." Journal de Théorie des Nombres de Bordeaux 20, no. 2 (2008): 365–371.
  • August 2005 (Revised April 2008)
  • Teaching Note

Store24 (B): Statistics Review with Data Desk (TN)

By: Frances X. Frei
Presents an overview of the statistical analysis covered in the case discussion. View Details
Keywords: Mathematical Methods; Retail Industry
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Frei, Frances X. "Store24 (B): Statistics Review with Data Desk (TN)." Harvard Business School Teaching Note 606-037, August 2005. (Revised April 2008.)
  • fall 1992
  • Article

Exploring the Limits of the Technology S-curve, Part 1: Component Technologies

By: Clayton M. Christensen
Keywords: Technology; Mathematical Methods
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Christensen, Clayton M. "Exploring the Limits of the Technology S-curve, Part 1: Component Technologies." Production and Operations Management 1 (fall 1992): 334–357.
  • 20 Mar 2001 - 21 Mar 2001
  • Conference Presentation

QAP: The Quadratic Assignment Procedure

By: William B. Simpson
Keywords: Mathematical Methods
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Simpson, William B. "QAP: The Quadratic Assignment Procedure." Paper presented at the North American Stata Users' Group Meeting, March 20–21, 2001.
  • 1979
  • Dissertation

Inference in Partially Identified Models

By: Dutch Leonard
Keywords: Mathematical Methods
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Leonard, Dutch. "Inference in Partially Identified Models." Diss., Harvard University, 1979.
  • Article

An Optimization-based Econometric Framework for the Evaluation of Monetary Policy

By: Julio J. Rotemberg and Michael Woodford
Keywords: Framework; Mathematical Methods; Money; Policy
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Rotemberg, Julio J., and Michael Woodford. "An Optimization-based Econometric Framework for the Evaluation of Monetary Policy." NBER Macroeconomics Annual 12 (1997): 297–344.
  • March 1989 (Revised April 1998)
  • Case

Marriott Corporation: The Cost of Capital (Abridged)

By: Richard S. Ruback
Gives students the opportunity to explore how a company uses the Capital Asset Pricing Model (CAPM) to compute the cost of capital for each of its divisions. The use of Weighted Average Cost of Capital (WACC) formula and the mechanics of applying it are stressed. View Details
Keywords: Cost of Capital; Mathematical Methods
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Ruback, Richard S. "Marriott Corporation: The Cost of Capital (Abridged)." Harvard Business School Case 289-047, March 1989. (Revised April 1998.)
  • November 2019
  • Article

Full Substitutability

By: John William Hatfield, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky and Alexander Westkamp
Various forms of substitutability are essential for establishing the existence of equilibria and other useful properties in diverse settings such as matching, auctions, and exchange economies with indivisible goods. We extend earlier models’ definitions of... View Details
Keywords: Substitutability; Mathematical Methods; Auctions; Market Design
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Hatfield, John William, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp. "Full Substitutability." Theoretical Economics 14, no. 4 (November 2019): 1535–1590.
  • November 2012
  • Article

The Variance of Non-Parametric Treatment Effect Estimators in the Presence of Clustering

By: Samuel G. Hanson and Adi Sunderam
Non-parametric estimators of treatment effects are often applied in settings where clustering may be important. We provide a general methodology for consistently estimating the variance of a large class of non-parametric estimators, including the simple matching... View Details
Keywords: Treatment Effects; Matching Estimators; Clustering; Applications and Software; Mathematical Methods
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Hanson, Samuel G., and Adi Sunderam. "The Variance of Non-Parametric Treatment Effect Estimators in the Presence of Clustering." Review of Economics and Statistics 94, no. 4 (November 2012). (Stata and Matlab Code Here.)
  • Article

The Shapley Value as a von Neumann-Morgenstern Utility

By: A. E. Roth
Keywords: Value; Mathematical Methods
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Roth, A. E. "The Shapley Value as a von Neumann-Morgenstern Utility." Econometrica 45, no. 3 (April 1977): 657–664.
  • June 1994
  • Article

Influence of Mathematical Models in Finance on Practice: Past, Present and Future

By: Robert C. Merton
Keywords: Mathematical Methods; Finance
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Merton, Robert C. "Influence of Mathematical Models in Finance on Practice: Past, Present and Future." Series A. Philosophical Transactions of the Royal Society of London, Series A, Physical Sciences and Engineering 347 (June 1994): 451–463. (Reprinted in Financial Practice and Education, spring 1995.)
  • 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
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Goh, Joel, Kian Guan Lim, Melvyn Sim, and Weina Zhang. "Portfolio Value-at-Risk Optimization for Asymmetrically Distributed Asset Returns." European Journal of Operational Research 221, no. 2 (September 1, 2012): 397–406.
  • 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
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Bell, David E. "Exposure and Hedging." Harvard Business School Background Note 186-036, July 1985. (Revised March 1994.)
  • 1976
  • Book

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

By: Michael A. Wheeler
Keywords: Society; Mathematical Methods; United States
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Wheeler, Michael A. Lies, Damn Lies and Statistics: The Manipulation of Public Opinion in America. New York, NY: W. W. Norton & Company, 1976.
  • 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
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Srinivas, Suraj, Sebastian Bordt, and Himabindu Lakkaraju. "Which Models Have Perceptually-Aligned Gradients? An Explanation via Off-Manifold Robustness." Advances in Neural Information Processing Systems (NeurIPS) (2023).
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