Skip to Main Content
HBS Home
  • About
  • Academic Programs
  • Alumni
  • Faculty & Research
  • Baker Library
  • Giving
  • Harvard Business Review
  • Initiatives
  • News
  • Recruit
  • Map / Directions
Faculty & Research
  • Faculty
  • Research
  • Featured Topics
  • Academic Units
  • …→
  • Harvard Business School→
  • Faculty & Research→
  • Research
    • Research
    • Publications
    • Global Research Centers
    • Case Development
    • Initiatives & Projects
    • Research Services
    • Seminars & Conferences
    →
  • Publications→

Publications

Publications

Filter Results : (140) Arrow Down
Filter Results : (140) Arrow Down Arrow Up

Show Results For

  • All HBS Web  (278)
    • Faculty Publications  (140)

    Show Results For

    • All HBS Web  (278)
      • Faculty Publications  (140)

      Mathematical Modeling Remove Mathematical Modeling →

      Page 1 of 140 Results →

      Are you looking for?

      → Search All HBS Web
      • 2022
      • Article

      Which Explanation Should I Choose? A Function Approximation Perspective to Characterizing Post hoc Explanations

      By: Tessa Han, Suraj Srinivas and Himabindu Lakkaraju
      A critical problem in the field of post hoc explainability is the lack of a common foundational goal among methods. For example, some methods are motivated by function approximation, some by game theoretic notions, and some by obtaining clean visualizations. This...  View Details
      Keywords: Mathematical Methods; Decision Choices and Conditions; Analytics and Data Science
      Citation
      Read Now
      Related
      Han, Tessa, Suraj Srinivas, and Himabindu Lakkaraju. "Which Explanation Should I Choose? A Function Approximation Perspective to Characterizing Post hoc Explanations." Advances in Neural Information Processing Systems (NeurIPS) (2022). (Best Paper Award, International Conference on Machine Learning (ICML) Workshop on Interpretable ML in Healthcare.)
      • August 2022
      • Article

      Contract Duration and the Costs of Market Transactions

      By: Alexander MacKay
      The optimal duration of a supply contract balances the costs of reselecting a supplier against the costs of being matched to an inefficient supplier when the contract lasts too long. I develop a structural model of contract duration that captures this tradeoff and...  View Details
      Keywords: Supply Contracts; Intermediate Goods; Switching Costs; Vertical Relationships; Transaction Costs; Contract Duration; Identification; Supply Chain; Cost; Contracts; Auctions; Mathematical Methods
      Citation
      SSRN
      Find at Harvard
      Register to Read
      Read Now
      Related
      MacKay, Alexander. "Contract Duration and the Costs of Market Transactions." American Economic Journal: Microeconomics 14, no. 3 (August 2022): 164–212.
      • 2022
      • Article

      Fairness via Explanation Quality: Evaluating Disparities in the Quality of Post hoc Explanations

      By: Jessica Dai, Sohini Upadhyay, Ulrich Aivodji, Stephen Bach and Himabindu Lakkaraju
      As post hoc explanation methods are increasingly being leveraged to explain complex models in high-stakes settings, it becomes critical to ensure that the quality of the resulting explanations is consistently high across all subgroups of a population. For instance, it...  View Details
      Keywords: Prejudice and Bias; Mathematical Methods; Research; Analytics and Data Science
      Citation
      Read Now
      Purchase
      Related
      Dai, Jessica, Sohini Upadhyay, Ulrich Aivodji, Stephen Bach, and Himabindu Lakkaraju. "Fairness via Explanation Quality: Evaluating Disparities in the Quality of Post hoc Explanations." Proceedings of the AAAI/ACM Conference on Artificial Intelligence, Ethics, and Society (2022): 203–214.
      • 2022
      • Article

      Exploring Counterfactual Explanations Through the Lens of Adversarial Examples: A Theoretical and Empirical Analysis.

      By: Martin Pawelczyk, Chirag Agarwal, Shalmali Joshi, Sohini Upadhyay and Himabindu Lakkaraju
      As machine learning (ML) models become more widely deployed in high-stakes applications, counterfactual explanations have emerged as key tools for providing actionable model explanations in practice. Despite the growing popularity of counterfactual explanations, a...  View Details
      Keywords: Machine Learning Models; Counterfactual Explanations; Adversarial Examples; Mathematical Methods
      Citation
      Read Now
      Related
      Pawelczyk, Martin, Chirag Agarwal, Shalmali Joshi, Sohini Upadhyay, and Himabindu Lakkaraju. "Exploring Counterfactual Explanations Through the Lens of Adversarial Examples: A Theoretical and Empirical Analysis." Proceedings of the International Conference on Artificial Intelligence and Statistics (AISTATS) 25th (2022).
      • 2022
      • Working Paper

      A Linear Panel Model with Heterogeneous Coefficients and Variation in Exposure

      By: Jesse M. Shapiro and Liyang Sun
      Linear panel models featuring unit and time fixed effects appear in many areas of empirical economics. An active literature studies the interpretation of the ordinary least squares estimator of the model, commonly called the two-way fixed effects (TWFE) estimator, in...  View Details
      Keywords: Econometric Models; Mathematical Methods
      Citation
      Find at Harvard
      Read Now
      Related
      Shapiro, Jesse M., and Liyang Sun. "A Linear Panel Model with Heterogeneous Coefficients and Variation in Exposure." NBER Working Paper Series, No. 29976, April 2022.
      • March 2022 (Revised July 2022)
      • Module Note

      Linear Regression

      By: Iavor I. Bojinov, Michael Parzen and Paul Hamilton
      This note provides an overview of linear regression for an introductory data science course. It begins with a discussion of correlation, and explains why correlation does not necessarily imply causation. The note then describes the method of least squares, and how to...  View Details
      Keywords: Data Science; Linear Regression; Mathematical Modeling; Mathematical Methods; Analytics and Data Science
      Citation
      Related
      Bojinov, Iavor I., Michael Parzen, and Paul Hamilton. "Linear Regression." Harvard Business School Module Note 622-100, March 2022. (Revised July 2022.)
      • March 2022 (Revised July 2022)
      • Module Note

      Statistical Inference

      By: Iavor I. Bojinov, Michael Parzen and Paul Hamilton
      This note provides an overview of statistical inference for an introductory data science course. First, the note discusses samples and populations. Next the note describes how to calculate confidence intervals for means and proportions. Then it walks through the logic...  View Details
      Keywords: Data Science; Statistics; Mathematical Modeling; Mathematical Methods; Analytics and Data Science
      Citation
      Purchase
      Related
      Bojinov, Iavor I., Michael Parzen, and Paul Hamilton. "Statistical Inference." Harvard Business School Module Note 622-099, March 2022. (Revised July 2022.)
      • March 2022
      • Article

      Estimating the Effectiveness of Permanent Price Reductions for Competing Products Using Multivariate Bayesian Structural Time Series Models

      By: Fiammetta Menchetti and Iavor Bojinov
      Researchers regularly use synthetic control methods for estimating causal effects when a sub-set of units receive a single persistent treatment, and the rest are unaffected by the change. In many applications, however, units not assigned to treatment are nevertheless...  View Details
      Keywords: Causal Inference; Partial Interference; Synthetic Controls; Bayesian Structural Time Series; Mathematical Methods
      Citation
      Find at Harvard
      Read Now
      Related
      Menchetti, Fiammetta, and Iavor Bojinov. "Estimating the Effectiveness of Permanent Price Reductions for Competing Products Using Multivariate Bayesian Structural Time Series Models." Annals of Applied Statistics 16, no. 1 (March 2022): 414–435.
      • March 2022
      • Article

      Sensitivity Analysis of Agent-based Models: A New Protocol

      By: Emanuele Borgonovo, Marco Pangallo, Jan Rivkin, Leonardo Rizzo and Nicolaj Siggelkow
      Agent-based models (ABMs) are increasingly used in the management sciences. Though useful, ABMs are often critiqued: it is hard to discern why they produce the results they do and whether other assumptions would yield similar results. To help researchers address such...  View Details
      Keywords: Agent-based Modeling; Sensitivity Analysis; Design Of Experiments; Total Order Sensitivity Indices; Organizations; Behavior; Decision Making; Mathematical Methods
      Citation
      Read Now
      Related
      Borgonovo, Emanuele, Marco Pangallo, Jan Rivkin, Leonardo Rizzo, and Nicolaj Siggelkow. "Sensitivity Analysis of Agent-based Models: A New Protocol." Computational and Mathematical Organization Theory 28, no. 1 (March 2022): 52–94.
      • 2022
      • Working Paper

      The Disagreement Problem in Explainable Machine Learning: A Practitioner's Perspective

      By: Satyapriya Krishna, Tessa Han, Alex Gu, Javin Pombra, Shahin Jabbari, Steven Wu and Himabindu Lakkaraju
      As various post hoc explanation methods are increasingly being leveraged to explain complex models in high-stakes settings, it becomes critical to develop a deeper understanding of if and when the explanations output by these methods disagree with each other, and how...  View Details
      Keywords: AI and Machine Learning; Analytics and Data Science; Mathematical Methods
      Citation
      Read Now
      Related
      Krishna, Satyapriya, Tessa Han, Alex Gu, Javin Pombra, Shahin Jabbari, Steven Wu, and Himabindu Lakkaraju. "The Disagreement Problem in Explainable Machine Learning: A Practitioner's Perspective." Working Paper, 2022.
      • September 2021
      • Article

      Diagnostic Bubbles

      By: Pedro Bordalo, Nicola Gennaioli, Spencer Yongwook Kwon and Andrei Shleifer
      We introduce diagnostic expectations into a standard setting of price formation in which investors learn about the fundamental value of an asset and trade it. We study the interaction of diagnostic expectations with two well-known mechanisms: learning from prices and...  View Details
      Keywords: Bubble; Speculation; Diagnostic Expectations; Price Bubble; Mathematical Methods
      Citation
      Find at Harvard
      Related
      Bordalo, Pedro, Nicola Gennaioli, Spencer Yongwook Kwon, and Andrei Shleifer. "Diagnostic Bubbles." Journal of Financial Economics 141, no. 3 (September 2021).
      • Article

      Learning Models for Actionable Recourse

      By: Alexis Ross, Himabindu Lakkaraju and Osbert Bastani
      As machine learning models are increasingly deployed in high-stakes domains such as legal and financial decision-making, there has been growing interest in post-hoc methods for generating counterfactual explanations. Such explanations provide individuals adversely...  View Details
      Keywords: Machine Learning Models; Recourse; Algorithm; Mathematical Methods
      Citation
      Read Now
      Related
      Ross, Alexis, Himabindu Lakkaraju, and Osbert Bastani. "Learning Models for Actionable Recourse." Advances in Neural Information Processing Systems (NeurIPS) 34 (2021).
      • June 2021
      • Technical Note

      Introduction to Linear Regression

      By: Michael Parzen and Paul Hamilton
      This technical note introduces (from an applied point of view) the theory and application of simple and multiple linear regression. The motivation for the model is introduced, as well as how to interpret the summary output with regard to prediction and statistical...  View Details
      Keywords: Linear Regression; Regression; Analysis; Forecasting and Prediction; Risk and Uncertainty; Theory; Compensation and Benefits; Mathematical Methods; Analytics and Data Science
      Citation
      Educators
      Related
      Parzen, Michael, and Paul Hamilton. "Introduction to Linear Regression." Harvard Business School Technical Note 621-086, June 2021.
      • April 2021
      • Article

      A Model of Multi-Pass Search: Price Search Across Stores and Time

      By: Navid Mojir and K. Sudhir
      In retail settings with price promotions, consumers often search across stores and time. However, the search literature typically only models one pass search across stores, ignoring revisits to stores; the choice literature using scanner data has modeled search across...  View Details
      Keywords: Consumer Search; Multi-pass Search; Price Search; Store Search; Spatial Search; Temporal Search; Spatiotemporal Search; Dynamic Structural Models; MPEC; Price Promotions; Store Loyalty; Consumer Behavior; Price; Spending; Marketing; Mathematical Methods
      Citation
      SSRN
      Find at Harvard
      Related
      Mojir, Navid, and K. Sudhir. "A Model of Multi-Pass Search: Price Search Across Stores and Time." Management Science 67, no. 4 (April 2021): 2126–2150.
      • February 2021
      • Tutorial

      What is AI?

      By: Tsedal Neeley
      This video explores the elements that constitute artificial intelligence (AI). From its mathematical basis to current advances in AI, this video introduces students to data, tools, and statistical models that make a computer 'intelligent.' Through an explanation of...  View Details
      Keywords: Artificial Intelligence; Digital; Technological Innovation; Leadership; AI and Machine Learning; Mathematical Methods
      Citation
      Purchase
      Related
      Neeley, Tsedal. What is AI? Harvard Business School Tutorial 421-713, February 2021. (https://hbsp.harvard.edu/product/421713-HTM-ENG?Ntt=tsedal%20neeley%20what%20is%20ai.)
      • February 2021
      • Article

      A Dynamic Theory of Multiple Borrowing

      By: Daniel Green and Ernest Liu
      Multiple borrowing—a borrower obtains overlapping loans from multiple lenders—is a common phenomenon in many credit markets. We build a highly tractable, dynamic model of multiple borrowing and show that, because overlapping creditors may impose default externalities...  View Details
      Keywords: Commitment; Multiple Borrowing; Common Agency; Misallocation; Microfinance; Investment; Mathematical Methods
      Citation
      SSRN
      Find at Harvard
      Related
      Green, Daniel, and Ernest Liu. "A Dynamic Theory of Multiple Borrowing." Journal of Financial Economics 139, no. 2 (February 2021): 389–404.
      • 2021
      • Article

      Fair Algorithms for Infinite and Contextual Bandits

      By: Matthew Joseph, Michael J Kearns, Jamie Morgenstern, Seth Neel and Aaron Leon Roth
      We study fairness in linear bandit problems. Starting from the notion of meritocratic fairness introduced in Joseph et al. [2016], we carry out a more refined analysis of a more general problem, achieving better performance guarantees with fewer modelling assumptions...  View Details
      Keywords: Algorithms; Bandit Problems; Fairness; Mathematical Methods
      Citation
      Read Now
      Related
      Joseph, Matthew, Michael J Kearns, Jamie Morgenstern, Seth Neel, and Aaron Leon Roth. "Fair Algorithms for Infinite and Contextual Bandits." Proceedings of the AAAI/ACM Conference on Artificial Intelligence, Ethics, and Society 4th (2021).
      • January 2021
      • Article

      Using Models to Persuade

      By: Joshua Schwartzstein and Adi Sunderam
      We present a framework where "model persuaders" influence receivers’ beliefs by proposing models that organize past data to make predictions. Receivers are assumed to find models more compelling when they better explain the data, fixing receivers’ prior beliefs. Model...  View Details
      Keywords: Model Persuasion; Analytics and Data Science; Forecasting and Prediction; Mathematical Methods; Framework
      Citation
      Find at Harvard
      Read Now
      Related
      Schwartzstein, Joshua, and Adi Sunderam. "Using Models to Persuade." American Economic Review 111, no. 1 (January 2021): 276–323.
      • Article

      Beyond Individualized Recourse: Interpretable and Interactive Summaries of Actionable Recourses

      By: Kaivalya Rawal and Himabindu Lakkaraju
      As predictive models are increasingly being deployed in high-stakes decision-making, there has been a lot of interest in developing algorithms which can provide recourses to affected individuals. While developing such tools is important, it is even more critical to...  View Details
      Keywords: Predictive Models; Decision Making; Framework; Mathematical Methods
      Citation
      Read Now
      Related
      Rawal, Kaivalya, and Himabindu Lakkaraju. "Beyond Individualized Recourse: Interpretable and Interactive Summaries of Actionable Recourses." Advances in Neural Information Processing Systems (NeurIPS) 33 (2020).
      • August 2020 (Revised September 2020)
      • Technical Note

      Assessing Prediction Accuracy of Machine Learning Models

      By: Michael W. Toffel, Natalie Epstein, Kris Ferreira and Yael Grushka-Cockayne
      The note introduces a variety of methods to assess the accuracy of machine learning prediction models. The note begins by briefly introducing machine learning, overfitting, training versus test datasets, and cross validation. The following accuracy metrics and tools...  View Details
      Keywords: Machine Learning; Statistics; Econometric Analyses; Experimental Methods; Data Analysis; Data Analytics; Forecasting and Prediction; Analytics and Data Science; Analysis; Mathematical Methods
      Citation
      Educators
      Purchase
      Related
      Toffel, Michael W., Natalie Epstein, Kris Ferreira, and Yael Grushka-Cockayne. "Assessing Prediction Accuracy of Machine Learning Models." Harvard Business School Technical Note 621-045, August 2020. (Revised September 2020.)
      • 1
      • 2
      • …
      • 6
      • 7
      • →

      Are you looking for?

      → Search All HBS Web
      ǁ
      Campus Map
      Harvard Business School
      Soldiers Field
      Boston, MA 02163
      →Map & Directions
      →More Contact Information
      • Make a Gift
      • Site Map
      • Jobs
      • Harvard University
      • Trademarks
      • Policies
      • Accessibility
      • Digital Accessibility
      Copyright © President & Fellows of Harvard College