# #Post-Selection #StatisticalInference in the era of #MachineLearning

#### Posted on May 6th, 2017

#datascience@columbia.edu

05/04/2017 @ ColumbiaUniversity, DavisAuditorium, CEPSR

Robert Tibshirani @StanfordUniversity talked about the adjusting the cutoffs for statistical significance testing of multiple null hypotheses. The #Bonferroni Correction has been used to adjustments for testing multiple hypothesis when the hypotheses are statistically independent. However, with the advent of #MachineLearning techniques, the number of possible tests and their interdependence has exploded.

This is especially true with the application of machine learning algorithms to large data sets with many possible independent variables which often use forward stepwise or Lasso regression procedures. Machine learning methods often use #regularization methods to avoid #overfitting the data such as data splitting into training, test and validation sets. For big data applications, these may be adequate since the emphasis on is prediction, not inference. Also the large size of the data set offsets issues such as the lower of power in the statistical tests conducted on a subset of the data.

Robert proposed a model for incremental variable selection in which each sequential test sliced off parts of the distribution for subsequent tests creating a truncated normal upon which one can assess the probability of the null hypothesis. This method of polyhedral selection works for a stepwise regression and well as a lasso regression with a fixed lambda.

When the value of lambda is determined by cross-validation, can use this method by adding 0.1 * sigma noise to the y values. This adjustment retains the power of the test and does not underestimate the probability of accepting the null hypothesis. This method can also be extended to other methods such as logistic regression, Cox proportional hazards model, graphics lasso.

The method can also be extended to consider the number of factors to use in the regression. This goals of this methodology are similar to those described by Bradley #Efron in his 2013 JASA paper on bootstrapping (http://statweb.stanford.edu/~ckirby/brad/papers/2013ModelSelection.pdf) and random matrix theory used to determine the number of principal components in the data as described by the #Marchenko-Pastur distribution.

There is a package in R: selectiveInference

Further information can be found in a chapter on ‘Statistical Learning with Sparsity’ by Hastie, Tibshirani, Wainwright (online pdf) and ‘Statistical Learning and selective inference’ (2015) Jonathan Taylor and Robert J. Tibshirani (PNAS)