Backcalculations is an important method for reconstructing past rates of human immunodeficiency virus (HIV) infection and for estimating current prevalence of HIV infection and future incidence of acquired immunodeficiency syndrome (AIDS). This paper reviews the backcalculation technique, focusing on the key assumptions of the method, including the necessary information regarding incubation, reporting delay, and models for the infection curve. A summary is given of the extend to which the appropriate external information is available and whether checks of the relevant assumptions are possible through use of data on AIDS incidence from surveillance systems. A likelihood approach to backcalculation is described and implemented on AIDS incidence data in the United States. New features of the approach influence incorporation of seasonal variation in diagnosis rates, smooth nonparametric estimation of both the HIV infection curve and nonstationary aspects of the incubation period and reporting delay distributions, and an analysis of residuals from backcalculation fits. Unexplained lack of fit is examined and discussed. A fundamental concern is the appropriate acknowledgment of uncertainty associated with backcalculation estimates caused by misspecified assumptions and inaccurate external estimates of key components of the technique. Such uncertainty limits the usefulness of backcalculation and highlights the need for complementary approaches.

A kernel smoother is an intuitive estimate of a regression function or conditional expectation; at each point $x_0$ the estimate of $E(Y\mid x_0)$ is a weighted mean of the sample $Y_i$, with observations close to $x_0$ receiving the largest weights. Unfortunately this simplicity has flaws. At the boundary of the predictor space, the kernel neighborhood is asymmetric and the estimate may have substantial bias. Bias can be a problem in the interior as well if the predictors are nonuniform or if the regression function has substantial curvature. These problems are particularly severe when the predictors are multidimensional. A variety of kernel modifications have been proposed to provide approximate and asymptotic adjustment for these biases. Such methods generally place substantial restrictions on the regression problems that can be considered; in unfavorable situations, they can perform very poorly. Moreover, the necessary modifications are very difficult to implement in the multidimensional case. Local regression smoothers fit lower-order polynomials in $x$ locally at $x_0$, and the estimate of $f(x_0)$ is taken from the fitted polynomial at $x_0$. They automatically, intuitively and simultaneously adjust for both the biases above to the given order and generalize naturally to the multidimensional case. They also provide natural estimates for the derivatives of $f$, an approach more attractive than using higher-order kernel functions for the same purpose.

It often appears that the most, indeed perhaps the only, effective role of statistics is to bolster decisions policymakers were prepared to take on other grounds. The secondary effects of smoking, the sex differential in SAT scores, the census undercount controvery, the validity of DNA evidence and the evidence of the relation of the race of the victim to the imposition of the death pernalty all provide examples of the intertwining of political and statistical considerations. Although the court in Alabama v. United States (1962) attributed to statistics the power to speak, if courts do not want to hear what the statistics are saying they just do not listen. Moreove, it sometimes appears that statistics are less likely to be relied upon if they challenge one's own interest. Confronted at their own institutions with an analyssis of faculty salaries showing evidence of discrimination, often the very faculty whose stock-in-trade is to persuade others of the efficacy of sttistics refuse to believe what is presented to them. Courts share this skepticism; this paper examines the courts's treatment of the statistical issues involved in studying possible discrimination in faculty salaries.

We examine a few key problems and their solutions from the 13 probability problems. Some are badly posed with imaginative but incorrect solutions; other are difficult, interesting and with correct solution. The work of Carroll (C. L. Dodgson) is used to illustrate the nature, standing and understanding of probability within the wider English mathematical community of his time. Additionally, a probabilistic controversy in which he was involved is discussed, and an Appendix presents a Markov chain formulation of published and unpublished problems and discussion of a further unpublished problem. One focus of the paper is the intuitive difficulty in distinguishing between events of probability zero and impossible events, and the teaching of such probability-based difficulties.