How to go data mining for models without “dredging” for models. (accidentally or otherwise) If you keep on testing models until you find some that fit (which you usually will) how do you know that the fit is in some sense *interesting*? How sharp will your conclusions be? How does it work when you are testing against a possibly uncountable continuum of hypotheses? (One perspective on sparsity penalties is precisely this, I am told.)

Model selection is this writ small - when you are testing how many variables to include in your model.

In modern high-dimensional models, where you have potentially many explanatory variables, the question of how to handle the combinatorial explosion of possible variables to include, this can also be considered a multiple testing problem. We tend to regard this as a smoothing and model selection problem though.

This all gets more complicated when you think about many people testing many hypothesese in many different experiments then you are going to run into many more issues than just these - also publication bias and suchlike.

Suggestive connection:

Moritz Hardt, The machine learning leaderboard problem:

In this post, I will describe a method to climb the public leaderboard without even looking at the data. The algorithm is so simple and natural that an unwitting analyst might just run it. We will see that in Kaggle’s famous Heritage Health Prize competition this might have propelled a participant from rank around 150 into the top 10 on the public leaderboard without making progress on the actual problem.[…]

I get super excited. I keep climbing the leaderboard! Who would’ve thought that this machine learning thing was so easy? So, I go write a blog post on Medium about Big Data and score a job at DeepCompeting.ly, the latest data science startup in the city. Life is pretty sweet. I pick up indoor rock climbing, sign up for wood working classes; I read Proust and books about espresso. Two months later the competition closes and Kaggle releases the final score. What an embarrassment! Wacky boosting did nothing whatsoever on the final test set. I get fired from DeepCompeting.ly days before the buyout. My spouse dumps me. The lease expires. I get evicted from my apartment in the Mission. Inevitably, I hike the Pacific Crest Trail and write a novel about it.

See BlHa15 and DFHP15 for more of that.

## P-value hacking

I Fooled Millions Into Thinking Chocolate Helps Weight Loss. Here’s How - also the journalism problem, the journals problem, the vacuous-fluff-that-passes-for-public-discussion problem…

## False discovery rate

FDR control…

- Testing Millions of Hypotheses is Larry Wasserman’s introduction to controlling the
*false discovery rate*. See also Screening and the false discovery rate. The man can explain clearly.

## Familywise error rate

Šidák correction, Bonferroni correction…

## Post selection inference

## Misc applied

http://kadavy.net/blog/posts/aa-testing/ http://businessofsoftware.org/2013/06/jason-cohen-ceo-wp-engine-why-data-can-make-you-do-the-wrong-thing/ http://www.evanmiller.org/the-low-base-rate-problem.html

# Refs

- Stor02: (2002) A direct approach to false discovery rates.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 64(3), 479–498. DOI - Tibs14: (2014) A General Framework for Fast Stagewise Algorithms.
*ArXiv:1408.5801 [Stat]*. - EfNi08: (2008) A method to compute multiplicity corrected confidence intervals for odds ratios and other relative effect estimates.
*International Journal of Environmental Research and Public Health*, 5(5), 394–398. - RGSG17: (2017) A million variables and more: the Fast Greedy Equivalence Search algorithm for learning high-dimensional graphical causal models, with an application to functional magnetic resonance images.
*International Journal of Data Science and Analytics*, 3(2), 121–129. DOI - FaLv10: (2010) A Selective Overview of Variable Selection in High Dimensional Feature Space.
*Statistica Sinica*, 20(1), 101–148. - LaMP15: (2015) A significance test for covariates in nonparametric regression.
*Electronic Journal of Statistics*, 9, 643–678. DOI - LTTT14: (2014) A significance test for the lasso.
*The Annals of Statistics*, 42(2), 413–468. DOI - BeGa09: (2009) A simple forward selection procedure based on false discovery rate control.
*The Annals of Applied Statistics*, 3(1), 179–198. DOI - DoJo95: (1995) Adapting to Unknown Smoothness via Wavelet Shrinkage.
*Journal of the American Statistical Association*, 90(432), 1200–1224. DOI - ABDJ06: (2006) Adapting to unknown sparsity by controlling the false discovery rate.
*The Annals of Statistics*, 34(2), 584–653. DOI - GeWa08: (2008) Adaptive confidence bands.
*The Annals of Statistics*, 36(2), 875–905. DOI - AiGe96: (1996) Adjusting for multiple testing when reporting research results: the Bonferroni vs Holm methods.
*American Journal of Public Health*, 86(5), 726–728. DOI - CaSh98: (1998) An Akaike information criterion for model selection in the presence of incomplete data.
*Journal of Statistical Planning and Inference*, 67(1), 45–65. DOI - Ston77: (1977) An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike’s Criterion.
*Journal of the Royal Statistical Society. Series B (Methodological)*, 39(1), 44–47. - ClKO09: (2009) Asymptotic properties of penalized spline estimators.
*Biometrika*, 96(3), 529–544. DOI - Dasg08: (2008)
*Asymptotic Theory of Statistics and Probability*. New York: Springer New York - BaHy01: (2001) Bandwidth selection for kernel conditional density estimation.
*Computational Statistics & Data Analysis*, 36(3), 279–298. DOI - Efro79: (1979) Bootstrap methods: another look at the jackknife.
*The Annals of Statistics*, 7(1), 1–26. DOI - RFFE15: (2015) Choosing experiments to accelerate collective discovery.
*Proceedings of the National Academy of Sciences*, 112(47), 14569–14574. DOI - ZhZh14: (2014) Confidence intervals for low dimensional parameters in high dimensional linear models.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 76(1), 217–242. DOI - EwSc15: (2015) Confidence Sets Based on the Lasso Estimator.
*ArXiv:1507.05315 [Math, Stat]*. - NiGe13: (2013) Confidence sets in sparse regression.
*The Annals of Statistics*, 41(6), 2852–2876. DOI - Bune04: (2004) Consistent covariate selection and post model selection inference in semiparametric regression.
*The Annals of Statistics*, 32(3), 898–927. DOI - BeHo95: (1995) Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.
*Journal of the Royal Statistical Society. Series B (Methodological)*, 57(1), 289–300. - BaCa15: (2015) Controlling the false discovery rate via knockoffs.
*The Annals of Statistics*, 43(5), 2055–2085. DOI - Efro10a: (2010a) Correlated z-values and the accuracy of large-scale statistical estimates.
*Journal of the American Statistical Association*, 105(491), 1042–1055. DOI - Küns86: (1986) Discrimination between monotonic trends and long-range dependence.
*Journal of Applied Probability*, 23(4), 1025–1030. - Efro07: (2007) Doing thousands of hypothesis tests at the same time.
*Metron - International Journal of Statistics*, LXV(1), 3–21. - JaFH15: (2015) Effective degrees of freedom: a flawed metaphor.
*Biometrika*, 102(2), 479–485. DOI - Efro09: (2009) Empirical Bayes Estimates for Large-Scale Prediction Problems.
*Journal of the American Statistical Association*, 104(487), 1015–1028. DOI - CaWB08: (2008) Enhancing Sparsity by Reweighted ℓ 1 Minimization.
*Journal of Fourier Analysis and Applications*, 14(5–6), 877–905. DOI - MeRi06: (2006) Estimating the proportion of false null hypotheses among a large number of independently tested hypotheses.
*The Annals of Statistics*, 34(1), 373–393. DOI - AnKo85: (1985) Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions.
*The Annals of Statistics*, 13(4), 1286–1316. DOI - TLTT14: (2014) Exact Post-selection Inference for Forward Stepwise and Least Angle Regression.
*ArXiv:1401.3889 [Stat]*. - LSST13: (2013) Exact post-selection inference, with application to the lasso.
*ArXiv:1311.6238 [Math, Stat]*. - SuBC15: (2015) False Discoveries Occur Early on the Lasso Path.
*ArXiv:1511.01957 [Cs, Math, Stat]*. - Mein06: (2006) False Discovery Control for Multiple Tests of Association Under General Dependence.
*Scandinavian Journal of Statistics*, 33(2), 227–237. DOI - TKPS14: (2014) False discovery rate smoothing.
*ArXiv:1411.6144 [Stat]*. - BeYe05: (2005) False Discovery Rate–Adjusted Multiple Confidence Intervals for Selected Parameters.
*Journal of the American Statistical Association*, 100(469), 71–81. DOI - KoKi96: (1996) Generalised information criteria in model selection.
*Biometrika*, 83(4), 875–890. DOI - Mein14: (2014) Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, n/a-n/a. DOI - MeBü06: (2006) High-dimensional graphs and variable selection with the lasso.
*The Annals of Statistics*, 34(3), 1436–1462. DOI - DBMM14: (2014) High-dimensional Inference: Confidence intervals, p-values and R-Software hdi.
*ArXiv:1408.4026 [Stat]*. - BüGe15: (2015) High-dimensional inference in misspecified linear models.
*ArXiv:1503.06426 [Stat]*, 9(1), 1449–1473. DOI - WaRo09: (2009) High-dimensional variable selection.
*Annals of Statistics*, 37(5A), 2178–2201. DOI - SiLi14: (2014) Higher Criticism: p-values and Criticism.
*ArXiv:1411.1437 [Math, Stat]*. - LSWA15: (2015) High-Reproducibility and High-Accuracy Method for Automated Topic Classification.
*Physical Review X*, 5(1), 011007. DOI - JaGe15: (2015) Honest confidence regions and optimality in high-dimensional precision matrix estimation.
*ArXiv:1507.02061 [Math, Stat]*. - Efro86: (1986) How biased is the apparent error rate of a prediction rule?
*Journal of the American Statistical Association*, 81(394), 461–470. DOI - Nobl09: (2009) How does multiple testing correction work?
*Nature Biotechnology*, 27(12), 1135–1137. DOI - CoBa17: (2017) Large numbers of explanatory variables, a semi-descriptive analysis.
*Proceedings of the National Academy of Sciences*, 114(32), 8592–8595. DOI - CaSu17: (2017) Large-Scale Global and Simultaneous Inference: Estimation and Testing in Very High Dimensions.
*Annual Review of Economics*, 9(1), 411–439. DOI - Efro13: (2013)
*Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction*. Cambridge: Cambridge University Press - MeYu09: (2009) Lasso-type recovery of sparse representations for high-dimensional data.
*The Annals of Statistics*, 37(1), 246–270. DOI - HCMF08: (2008) Least angle and ℓ1 penalized regression: A review.
*Statistics Surveys*, 2, 61–93. DOI - HjJo96: (1996) Locally parametric nonparametric density estimation.
*The Annals of Statistics*, 24(4), 1619–1647. DOI - MeBü05: (2005) Lower bounds for the number of false null hypotheses for multiple testing of associations under general dependence structures.
*Biometrika*, 92(4), 893–907. DOI - GoWh04: (2004) Maximum likelihood and the bootstrap for nonlinear dynamic models.
*Journal of Econometrics*, 119(1), 199–219. DOI - GLAB14: (2014) Metric Learning for Temporal Sequence Alignment. In Advances in Neural Information Processing Systems 27 (pp. 1817–1825). Curran Associates, Inc.
- BuBA97: (1997) Model Selection: An Integral Part of Inference.
*Biometrics*, 53(2), 603–618. DOI - Bach09: (2009) Model-Consistent Sparse Estimation through the Bootstrap
- BuAn04: (2004) Multimodel Inference Understanding AIC and BIC in Model Selection.
*Sociological Methods & Research*, 33(2), 261–304. DOI - Roth90: (1990) No adjustments are needed for multiple comparisons.
*Epidemiology (Cambridge, Mass.)*, 1(1), 43–46. - ArEm11: (2011) Nonparametric goodness-of-fit tests for discrete null distributions.
*The R Journal*, 3(2), 34–39. - GBRD14: (2014) On asymptotically optimal confidence regions and tests for high-dimensional models.
*The Annals of Statistics*, 42(3), 1166–1202. DOI - DeHM08: (2008) On Deconvolution with Repeated Measurements.
*The Annals of Statistics*, 36(2), 665–685. DOI - Hjor92: (1992) On Inference in Parametric Survival Data Models.
*International Statistical Review / Revue Internationale de Statistique*, 60(3), 355–387. DOI - ZoHT07: (2007) On the “degrees of freedom” of the lasso.
*The Annals of Statistics*, 35(5), 2173–2192. DOI - Tadd13: (2013) One-step estimator paths for concave regularization.
*ArXiv:1308.5623 [Stat]*. - CFJL16: (2016) Panning for Gold: Model-free Knockoffs for High-dimensional Controlled Variable Selection.
*ArXiv Preprint ArXiv:1610.02351*. - RoZh07: (2007) Piecewise linear regularized solution paths.
*The Annals of Statistics*, 35(3), 1012–1030. DOI - DFHP14: (2014) Preserving Statistical Validity in Adaptive Data Analysis.
*ArXiv:1411.2664 [Cs]*. - MeMB09: (2009) p-Values for High-Dimensional Regression.
*Journal of the American Statistical Association*, 104(488), 1671–1681. DOI - MüBe14: (2014) pystruct - Learning Structured Prediction in Python.
*Journal of Machine Learning Research*, 15, 2055–2060. - EvDi00: (n.d.) Recovering from Selection Bias using Marginal Structure in Discrete Models.
- HuTs89: (1989) Regression and time series model selection in small samples.
*Biometrika*, 76(2), 297–307. DOI - FrHT10: (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent.
*Journal of Statistical Software*, 33(1), 1–22. DOI - Mein07: (2007) Relaxed Lasso.
*Computational Statistics & Data Analysis*, 52(1), 374–393. DOI - CaRT06: (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information.
*IEEE Transactions on Information Theory*, 52(2), 489–509. DOI - Efro04a: (2004a) Selection and Estimation for Large-Scale Simultaneous Inference.
- IyGr88: (1988) Selection Models and the File Drawer Problem.
*Statistical Science*, 3(1), 109–117. DOI - HjWL92: (1992) Semiparametric Estimation Of Parametric Hazard Rates. In Survival Analysis: State of the Art (pp. 211–236). Springer Netherlands
- Ichi93: (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models.
*Journal of Econometrics*, 58(1–2), 71–120. DOI - KoCW15: (2015) Sequential Tests for Large-Scale Learning.
*Neural Computation*, 28(1), 45–70. DOI - CuVD11: (2011) Significance testing in ridge regression for genetic data.
*BMC Bioinformatics*, 12, 372. DOI - Benj10: (2010) Simultaneous and selective inference: Current successes and future challenges.
*Biometrical Journal*, 52(6), 708–721. DOI - ClZi75: (1975) Simultaneous Estimation of the Means of Independent Poisson Laws.
*Journal of the American Statistical Association*, 70(351a), 698–705. DOI - Efro08: (2008) Simultaneous Inference: When Should Hypothesis Testing Problems Be Combined?
*The Annals of Applied Statistics*, 2(1), 197–223. DOI - MeBü10: (2010) Stability selection.
*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*, 72(4), 417–473. DOI - CoMa85: (1985) Testing Goodness of Fit for the Poisson Assumption When Observations Are Not Identically Distributed.
*Journal of the American Statistical Association*, 80(390), 411–418. DOI - LaRa12: (2012) The cost of large numbers of hypothesis tests on power, effect size and sample size.
*Molecular Psychiatry*, 17(1), 108–114. DOI - Efro04b: (2004b) The Estimation of Prediction Error.
*Journal of the American Statistical Association*, 99(467), 619–632. DOI - Efro10b: (2010b) The Future of Indirect Evidence.
*Statistical Science*, 25(2), 145–157. DOI - DaBa16: (2016) The knockoff filter for FDR control in group-sparse and multitask regression.
*ArXiv Preprint ArXiv:1602.03589*. - BlHa15: (2015) The Ladder: A Reliable Leaderboard for Machine Learning Competitions.
*ArXiv:1502.04585 [Cs]*. - GeLe11: (2011) The Lasso, correlated design, and improved oracle inequalities.
*ArXiv:1107.0189 [Stat]*. - DFHP15: (2015) The reusable holdout: Preserving validity in adaptive data analysis.
*Science*, 349(6248), 636–638. DOI - GeLo14: (2014) The Statistical Crisis in Science.
*American Scientist*, 102(6), 460. DOI - FrLu14: (2014) Unconscious lie detection as an example of a widespread fallacy in the Neurosciences.
*ArXiv:1407.4240 [q-Bio, Stat]*. - TRTW15: (2015) Uniform Asymptotic Inference and the Bootstrap After Model Selection.
*ArXiv:1506.06266 [Math, Stat]*. - Cava97: (1997) Unifying the derivations for the Akaike and corrected Akaike information criteria.
*Statistics & Probability Letters*, 33(2), 201–208. DOI - ChHS15: (2015) Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach.
*Annual Review of Economics*, 7(1), 649–688. DOI - BBBZ13: (2013) Valid post-selection inference.
*The Annals of Statistics*, 41(2), 802–837. DOI - LiLi08: (2008) Variable selection in semiparametric regression modeling.
*The Annals of Statistics*, 36(1), 261–286. DOI - FaLi01: (2001) Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties.
*Journal of the American Statistical Association*, 96(456), 1348–1360. DOI - TPSR15: (2015) Vector-Space Markov Random Fields via Exponential Families. In Journal of Machine Learning Research (pp. 684–692).
- DJKP95: (1995) Wavelet Shrinkage: Asymptopia?
*Journal of the Royal Statistical Society. Series B (Methodological)*, 57(2), 301–369. - KaRo14: (2014) When does more regularization imply fewer degrees of freedom? Sufficient conditions and counterexamples.
*Biometrika*, 101(4), 771–784. DOI - Ioan05: (2005) Why most published research findings are false.
*PLoS Medicine*, 2(8), 124. DOI