---
author:
- |
    [**Hossein Soleimani[^1]**]{}\
    Computer Science Dept.\
    Johns Hopkins University\
    Baltimore, MD 21218\
    \
    Computer Science Dept.\
    Johns Hopkins University\
    Baltimore, MD 21218\
    \
    Computer Science Dept.\
    Johns Hopkins University\
    Baltimore, MD 21218\
title: 'Treatment-Response Models for Counterfactual Reasoning with Continuous-time, Continuous-valued Interventions'
---

INTRODUCTION {#intro_section}
============

RELATED WORK {#relwork_section}
============

METHODS {#method_section}
=======

EXPERIMENTS {#exp_section}
===========

CONCLUSION {#conc_section}
==========

### Acknowledgements {#acknowledgements .unnumbered}

The authors would like to thank Yanbo Xu for valuable discussions. 

Abadi, M., Agarwal, A., Barham, P., et al. ( 2014). .  Technical report. 

lvarez, M. A. and Lawrence, N. D. (2009). . ,  12:1459–1500. 

Bang, H. and Robins, J. M. (2005).  Doubly robust estimation in missing data and causal inference models. ,  61(4):962–973. 

Bottou, L., Peters, J., Qui[ñ]{}onero-Candela, J., et al. ( 2013).  Counterfactual reasoning and learning systems: The example of computational advertising. ,  14(1):3207–3260. 

Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C. (1995).  A limited memory algorithm for bound constrained optimization. ,  16(5):1190–1208. 

Card, D. (1999).  The causal effect of education on earnings. ,  3:1801–1863. 

Chipman, H. A., George, E. I., McCulloch, R. E., et al. ( 2010).  BART: Bayesian additive regression trees. ,  4(1):266–298. 

Chou, K.-J., Lee, P.-T., Chen, C.-L., et al. ( 2006).  Physiological changes during hemodialysis in patients with intradialysis hypertension. ,  69(10):1833–1838. 

Cutler, D. J. (1978).  Linear systems analysis in pharmacokinetics. ,  6(3):265–282. 

Ghassemi, M., Pimentel, M. A., Naumann, T., et al. ( 2015).  A multivariate timeseries modeling approach to severity of illness assessment and forecasting in icu with sparse, heterogeneous clinical data.  In [*AAAI*]{}, pages 446–453. 

Gill, R. D. and Robins, J. M. (2001).  Causal inference for complex longitudinal data: the continuous case. ,  pages 1785–1811. 

Golnaraghi, F. and Kuo, B. (2010).  Automatic control systems. ,  2:1–1. 

Green, D. P. and Kern, H. L. (2012).  Modeling heterogeneous treatment effects in survey experiments with Bayesian additive regression trees. ,  pages 491––511. 

Greenland, S. (1995).  Dose-response and trend analysis in epidemiology: alternatives to categorical analysis. ,  pages 356–365. 

Hensman, J., de G Matthews, A. G., et al. ( 2016). . . 

Hensman, J., Fusi, N., and Lawrence, N. D. (2013). .  In [*UAI*]{}. 

Hensman, J. and Matthews, A. (2015). .  In [*AISTATS*]{}, volume 38, pages 1–9. 

Hill, J. L. (2011).  Bayesian nonparametric modeling for causal inference. ,  20(1):217–240. 

Hochreiter, S. and Schmidhuber, J. (1997).  Long short-term memory. ,  9(8):1735–1780. 

Hong, G. and Yu, B. (2008).  Effects of kindergarten retention on children’s social-emotional development: an application of propensity score method to multivariate, multilevel data. ,  44(2):407. 

Johnson, B. A. and Tsiatis, A. A. (2005).  Semiparametric inference in observational duration-response studies, with duration possibly right-censored. ,  92(3):605–618. 

Li, L., Chen, S., Kleban, J., and Gupta, A. (2015).  Counterfactual estimation and optimization of click metrics in search engines: A case study.  In [*Int.  Conf.  on WWWW*]{}, pages 929–934.  ACM. 

Lipton, Z. C., Kale, D. C., and Wetzel, R. (2016). .  In [*Machine Learning for Healthcare*]{}, pages 1–17. 

Liu, Z. and Hauskrecht, M. (2016). .  In [*AAAI*]{}. 

Lok, J. J. (2008).  Statistical modeling of causal effects in continuous time. ,  pages 1464–1507. 

Moodie, E. E. and Stephens, D. A. (2012).  Estimation of dose–response functions for longitudinal data using the generalised propensity score. ,  21(2):149–166. 

Pannu, N. and Gibney, R. N. (2005).  Renal replacement therapy in the intensive care unit. ,  1(2):141. 

Pearl, J. et al. ( 2009).  Causal inference in statistics: An overview. ,  3:96–146. 

Rasmussen, C. E. and Williams, C. (2006). .  MIT Press. 

Rich, B., Moodie, E. E., and Stephens, D. A. (2016).  Optimal individualized dosing strategies: A pharmacologic approach to developing dynamic treatment regimens for continuous-valued treatments. ,  58(3):502–517. 

Robins, J. (1987).  A graphical approach to the identification and estimation of causal parameters in mortality studies with sustained exposure periods. ,  40:139S–161S.

Robins, J. M., Hernan, M. A., and Brumback, B. (2000).  Marginal structural models and causal inference in epidemiology. 

Rosenbaum, P. R. and Rubin, D. B. (1984).  Reducing bias in observational studies using subclassification on the propensity score. ,  79(387):516–524. 

Rubin, D. B. (1974).  Estimating causal effects of treatments in randomized and nonrandomized studies. ,  66(5):688. 

Rubin, D. B. (1976).  Inference and missing data. ,  63(3):581–592. 

Saeed, M., Lieu, C., Raber, G., and Mark, R. G. (2002).  Mimic II: a massive temporal ICU patient database to support research in intelligent patient monitoring.  In [*Computers in Cardiology, 2002*]{}, pages 641–644. 

Schnabel, T., Swaminathan, A., Singh, A., Chandak, N., and Joachims, T. (2016).  Recommendations as treatments: Debiasing learning and evaluation. . 

Schulam, P. and Saria, S. (2016).  Integrative analysis using coupled latent variable models for individualizing prognoses. ,  17(234):1–35. 

Schulam, P. and Saria, S. (2017).  What-if reasoning with counterfactual Gaussian processes. . 

Seeger, M., Teh, Y.-W., and Jordan, M. (2005).  Semiparametric latent factor models.  Technical report. 

Shargel, L., Andrew, B., and Wu-Pong, S. (2005). ,  volume 270.  McGraw-Hill New York:. 

Silva, R. (2016).  Observational-interventional priors for dose-response learning.  In [*Advances in Neural Information Processing Systems*]{}, pages 1561–1569. 

Tao, Y. (2016). .  PhD thesis, The University of Michigan. 

Titsias, M. K. (2009). .  In [*AISTATS*]{}. 

Verbeke, G. and Molenberghs, G. (2009). .  Springer Science & Business Media. 

Westreich, D., Cole, S. R., Young, J. G., et al. ( 2012).  The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident aids or death. ,  31(18):2000–2009. 

Wilke, R., Xu, H., Denny, J., Roden, D., Krauss, R., and otherwise (2011).  The emerging role of electronic medical records in pharmacogenomics. ,  89(3):379–386. 

Xu, Y., Xu, Y., and Saria, S. (2016).  A Bayesian nonparametic approach for estimating individualized treatment-response curves.  In [*Machine Learning for Healthcare*]{}, pages 282–300. 

Appendix {#append_section}
========

[^1]:   These authors contributed equally to this work.