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Fisherian alternatives to conventional statistics

22 August 2010 Leave a comment

Novel developments in statistics and information theory call for a reconsideration of important aspects of two of R. A. Fisher’s most controversial ideas: the fiducial argument and the direct use of the likelihood function. Some key features of observed confidence levels, the direct use of the likelihood function, and the minimum description length principle are summarized here:

  1. Like the fiducial distribution, a probability measure of observed confidence levels is in effect a posterior probability distribution of the parameter of interest that does not require any prior distribution. Derived from sets of confidence intervals, this probability distribution of a parameter of interest is traditionally known as a confidence distribution. When the parameter of interest is scalar, the observed confidence level of a composite hypothesis is equal to its fiducial probability. On the other hand, observed conference levels do not suffer from the difficulties of constructing a fiducial distribution of a vector parameter.
  2. The likelihood ratio serves not only as a tool for the construction of point estimators, p-values, confidence intervals, and posterior probabilities, but is also fruitfully interpreted as a measure of the strength of statistical evidence for one hypothesis over another through the lens of a family of distributions. Modern versions of Fisher’s evidential use of the likelihood overcome multiplicity problems that arise in standard frequentism without resorting to a prior distribution.
  3. A related approach is to select the family of distributions using a modern information-theoretic reinterpretation of the likelihood function. In particular, the minimum description length principle extends the scope of Fisherian likelihood inference to the challenging problem of model selection.
Categories: Fragments, MDL

Medium-scale simultaneous inference

14 August 2010 3 comments

D. R. Bickel, “Minimum description length methods of medium-scale simultaneous inference,” Technical Report, Ottawa Institute of Systems Biology, available at tinyurl.com/36dm6lj (2010). Full preprint

Abstract— Nonparametric statistical methods developed for analyzing data for high numbers of genes, SNPs, or other biological features tend to have low efficiency for data with the smaller numbers of features such as proteins, metabolites, or, when expression is measured with conventional instruments, genes. For this medium-scale inference problem, the minimum description length (MDL) framework quantifies the amount of information in the data supporting a null or alternative hypothesis for each feature in terms of parametric model selection. Two new MDL techniques are proposed. First, using test statistics that are highly informative about the parameter of interest, the data are reduced to a single statistic per feature. This simplifying step is already implicit in conventional hypothesis testing and has been found effective in empirical Bayes applications to genomics data. Second, the codelength difference between the alternative and null hypotheses of any given feature can take advantage of information in the measurements from all other features by using those measurements to find the overall code of minimum length summed over those features. The techniques are applied to protein abundance data, demonstrating that a computationally efficient approximation that is close for a sufficiently large number of features works well even when the number of features is as low as 20.

Keywords: information criteria; minimum description length; model selection; reduced likelihood

Information Theoretic Methods

11 August 2010 Leave a comment
Categories: Fragments, MDL