Spinoza on method and freedom

The thesis as a whole argues that Spinoza’s Ethics in both method and content is aimed at the normal, partly rational person. Chapter 1 is on Spinoza’s writing style, finding that rather than being arid and technical, it aims to convince the reader by means of various rhetorical techniques, so does not assume an already rational reader. The following chapters of Part 1 examine whether the Ethics’ use of the synthetic geometric method exposes it to Descartes’ critique of that method in the “Second Replies” to his Meditations, that it is not suitable for pedagogy. This involves a consideration of the role of the TIE, finding in that early text not the analytic wing of a two-part analytic-synthetic method, but rather a defence and necessitation of a stand-alone synthetic method. Part 2 of the thesis develops this study of Spinoza’s writing for the common man to consider whether he is writing about the common man. This is done by examining one of the seemingly most abstract propositions in the Ethics, 4P72, which claims that a free man will not deceive even to save his own life. The study examines who exactly is this “free man” and what is his role in the Ethics. The study looks at the examples of free men in the TTP and at the concept of the model in the Ethics, and finds that rather than the free man being an impossible ideal which we can aim at but never achieve, everyone is free to some extent, and that even normal people are at times “the free man”.

On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.