Statistical Hypothesis Testing in Positive Unlabelled Data
We propose a set of novel methodologies which enable valid statistical hypothesis testing when we have only positive and unlabelled (PU) examples. This type of problem, a special case of semi-supervised data, is common in text mining, bioinformatics, and computer vision. Focusing on a generalised likelihood ratio test, we have 3 key contributions: (1) a proof that assuming all unlabelled examples are negative cases is sufficient for independence testing, but not for power analysis activities; (2) a new methodology that compensates this and enables power analysis, allowing sample size determination for observing an effect with a desired power; and finally, (3) a new capability, supervision determination, which can determine a-priori the number of labelled examples the user must collect before being able to observe a desired statistical effect. Beyond general hypothesis testing, we suggest the tools will additionally be useful for information theoretic feature selection, and Bayesian Network structure learning.
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- 1.Agresti, A.: Categorical Data Analysis, 3rd edn. Wiley Series in Probability and Statistics. Wiley-Interscience (2013)Google Scholar
- 3.Blanchard, G., Lee, G., Scott, C.: Semi-Supervised Novelty Detection. Jour. of Mach. Learn. Res. 11 (March 2010)Google Scholar
- 6.Cohen, J.: Statistical Power Analysis for the Behavioral Sciences, 2nd edn. Routledge Academic (1988)Google Scholar
- 7.Cohn, D., Atlas, L., Ladner, R.: Improving generalization with active learning. Machine Learning 15(2), 201–221 (1994)Google Scholar
- 9.Denis, F., Laurent, A., Gilleron, R., Tommasi, M.: Text classification and co-training from positive and unlabeled examples. In: International Conf. on Machine Learning, Workshop: The Continuum from Labeled to Unlabeled Data (2003)Google Scholar
- 10.Elkan, C., Noto, K.: Learning classifiers from only positive and unlabeled data. In: SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (2008)Google Scholar
- 11.Ellis, P.: The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results. Camb. Univ. Press (2010)Google Scholar
- 12.Gretton, A., Györfi, L.: Consistent nonparametric tests of independence. The Journal of Machine Learning Research 99, 1391–1423 (2010)Google Scholar
- 14.Hahn, G., Shapiro, S.: Statistical Models in Engineering. Wiley Series on Systems Engineering and Analysis Series. John Wiley & Sons (1967)Google Scholar
- 15.Liu, B., Lee, W., Yu, P., Li, X.: Partially supervised classification of text documents. In: International Conf. on Machine Learning, pp. 387–394 (2002)Google Scholar
- 18.Sokal, R., Rohlf, F.: Biometry: The principles and practice of Statistics in Biological data, 3rd edn. W. H. Freeman & Co (1995)Google Scholar
- 19.Yu, H., Han, J., Chang, K.: PEBL: positive example based learning for web page classification using svm. In: SIGKDD Int. Conf. on Knowledge Discovery and Data Mining (2002)Google Scholar