Bayesian Spectrum Analysis and Parameter Estimation This book is primarily a research document on the application of probability theory to the parameter estimation problem. The people who will be interested in this material are physicists chemists economists and engineers who have to deal with data on a daily basis; consequently we have included a great deal of introductory and tutorial material. Any person with the equivalent of the mathematics background required for the graduate level study of physics should be able to follow the material contained in this book though not without effort. In this work we apply probability theory to the problem of estimating parameters in rather general models. In particular when the model consists of a single stationary sinusoid we show that the direct application of probability theory will yield frequency estimates an order of magnitude better than a discrete Fourier transform in signal to noise of one. Latter we generalize the problem and show that probability theory can separate two close frequencies long after the peaks in a discrete Fourier transform have merged