Herman Chernoff

The Use of Faces to Represent Points in k-Dimensional Space Graphically

Abstract A novel method of representing multivariate data is presented. Each point in k-dimensional space, k≤18, is represented by a cartoon of a face whose features, such as length of nose and curvature of mouth, correspond to components of the point. Thus every multivariate observation is visualized as a computer-drawn face. This presentation makes it easy for the human mind to grasp many of the essential regularities and irregularities present in the data. Other graphical representations are described briefly.

Sequential analysis and optimal design

Preliminaries on probability Generalities about the conventional theory of design of experiments Optimal sample size Preliminaries on regression Design for linear regression: Elfvings method Maximum-likelihood estimation Locally optimal designs for estimation More design in regression experiments Testing hypotheses Optimal sample size in testing Sequential probability-ratio test Optimality of sequential probability-ratio test Motivation for an approach to sequential design of experiments in testing hypotheses Asymptotic optimality of procedure A in sequential design Extensions and open questions in sequential design The problem of adjacent hypotheses Testing for the sign of a normal mean: no indifference zone Bandit problems Sequential estimation of a normal mean sequential estimation of the common mean of two normal populations.

Sequential Design of Experiments

Considerable scientific research is characterized as follows. The scientist is interested in studying a phenomenon. At first he is quite ignorant and his initial experiments are preliminary and tentative. As he gathers relevant data, he becomes more definite in his impression of the underlying theory. This more definite impression is used to construct more informative experiments. Finally after a certain point he is satisfied that his evidence is sufficient to allow him to announce certain conclusions and he does so.

Asymptotic Distribution of the Likelihood Ratio Test That a Mixture of Two Binomials is a Single Binomial

Abstract A problem of interest in genetics is that of testing whether a mixture of two binomial distributions Bi(k, p) and B i (k, 1 2 ) is simply the pure distribution B i (k, 1 2 ) . This problem arises in determining whether we have a genetic marker for a gene responsible for a heterogeneous trait, that is a trait which is caused by any one of several genes. In that event we would have a nontrivial mixture involving 0 Standard asymptotic theory breaks down for such problems which belong to a class of problems where a natural parametrization represents a single distribution, under the hypothesis to be tested, by infinitely many possible parameter points. That difficulty may be eliminated by a transformation of parameters. But in that case a second problem appears. The regularity conditions demanded by the applicability of the Fisher Information fails when k > 2. We present an approach where use is made of the Kullback Leibler information, of which the Fisher information is a limiting case. Several versions of the binomial mixture problem are studied. The asymptotic analysis is supplemented by the results of simulations. It is shown that as n → ∞, the asymptotic distribution of twice the logarithm of the likelihood ratio corresponds to the square of the supremum of a Gaussian stochastic process with mean 0, variance 1 and a well behaved covariance function. As k → ∞ this limiting distribution grows stochastically as log k.

Optimal Accelerated Life Designs for Estimation

A technique is developed to obtain optimal accelerated life designs for estimating the parameters describing the mean lifetime of a device under a standard environment. Five examples involving variations of models and experimental design are studied. An exponential distribution of lifetimes is assumed in each of the five problems. The general approach, applyin g a method of Elfving, is applicable to a large variety of models.

Effect on Classification Error of Random Permutations of Features in Representing Multivariate Data by Faces

Abstract A graphical method of representing multivariate data consists of drawing a cartoon of a face determined by 18 parameters. A sample of vector observations of dimension d ≤ 18 is converted to faces by assigning components of the vector to facial parameters. We report an experiment which estimates that the effect of a random permutation in the assignment of parameters may affect the error rate in a classification task using these faces by a factor of about 25 percent.

Personal probabilities of probabilities

By definition, the subjective probability distribution of a random event is revealed by the (‘rational’) subjects choice between bets — a view expressed by F. Ramsey, B. De Finetti, L. J. Savage and traceable to E. Borel and, it can be argued, to T. Bayes. Since hypotheses are not observable events, no bet can be made, and paid off, on a hypothesis. The subjective probability distribution of hypotheses (or of a parameter, as in the current ‘Bayesian’ statistical literature) is therefore a figure of speech, an ‘as if’, justifiable in the limit. Given a long sequence of previous observations, the subjective posterior probabilities of events still to be observed are derived by using a mathematical expression that would approximate the subjective probability distribution of hypotheses, if these could be bet on. This position was taken by most, but not all, respondents to a ‘Round Robin’ initiated by J. Marschak after M. H. De-Groots talk on Stopping Rules presented at the UCLA Interdisciplinary Colloquium on Mathematics in Behavioral Sciences. Other participants: K. Borch, H. Chernoif, R. Dorfman, W. Edwards, T. S. Ferguson, G. Graves, K. Miyasawa, P. Randolph, L. J. Savage, R. Schlaifer, R. L. Winkler. Attention is also drawn to K. Borchs article in this issue.

Why significant variables aren’t automatically good predictors

Significance A recent puzzle in the big data scientific literature is that an increase in explanatory variables found to be significantly correlated with an outcome variable does not necessarily lead to improvements in prediction. This problem occurs in both simple and complex data. We offer explanations and statistical insights into why higher significance does not automatically imply stronger predictivity and why variables with strong predictivity sometimes fail to be significant. We suggest shifting the research agenda toward searching for a criterion to locate highly predictive variables rather than highly significant variables. We offer an alternative approach, the partition retention method, which was effective in reducing prediction error from 30% to 8% on a long-studied breast cancer data set. Thus far, genome-wide association studies (GWAS) have been disappointing in the inability of investigators to use the results of identified, statistically significant variants in complex diseases to make predictions useful for personalized medicine. Why are significant variables not leading to good prediction of outcomes? We point out that this problem is prevalent in simple as well as complex data, in the sciences as well as the social sciences. We offer a brief explanation and some statistical insights on why higher significance cannot automatically imply stronger predictivity and illustrate through simulations and a real breast cancer example. We also demonstrate that highly predictive variables do not necessarily appear as highly significant, thus evading the researcher using significance-based methods. We point out that what makes variables good for prediction versus significance depends on different properties of the underlying distributions. If prediction is the goal, we must lay aside significance as the only selection standard. We suggest that progress in prediction requires efforts toward a new research agenda of searching for a novel criterion to retrieve highly predictive variables rather than highly significant variables. We offer an alternative approach that was not designed for significance, the partition retention method, which was very effective predicting on a long-studied breast cancer data set, by reducing the classification error rate from 30% to 8%.

Use of Normal Probability Paper

Abstract Normal probability paper is so designed that the cumulative distribution function of a normally distributed chance variable appears as a straight line. It is a common practice to plot the observations of a sample on this paper to obtain a graphical check for normality or to obtain a graphical estimate of the mean and variance of the population. Textbooks, however, are not very specific about methods for plotting, for, although the ordered observations are plotted along the abscissa, some uncertainties about the corresponding ordinates are left unresolved. The purpose of this paper is to indicate, with a special example, that any graphical technique should depend to a large extent on the purpose for which the graph is drawn. In particular, it presents tables covering sample sizes up to 10, for selecting the ordinates on normal probability paper so as to obtain “optimum” graphical estimates of the mean ζ and the standard deviation σ of a normal distribution. The somewhat more complicated problem of...

Numerical solutions for Bayes sequential decision problems

Certain sequential decision problems involving normal random variables reduce to optimal stopping problems which can be related to the solution of corresponding free boundary problems for the heat equation. The numerical solution of these free boundary problems can then be approximated by calculating the solution of simpler optimal stopping problems by backward induction. This approach is not well adapted for very precise results but is surprisingly effective for rough pproximations. An estimate of the difference between the solutions of the related problems permits one to make continuity corrections which provide considerably improved accuracy. Further reductions in the necessary computational effort are possible by considering truncated procedures for one-sided boundaries and by exploiting monotone and symmetric boundaries.