Bayesian Spectrum Analysis and Parameter Estimation (Lecture by G. Larry Bretthorst

By G. Larry Bretthorst

This paintings is largely an intensive revision of my Ph.D. dissertation, [1J. It 1S basically a examine rfile at the program of chance idea to the parameter estimation challenge. the folks who could be attracted to this fabric are physicists, economists, and engineers who've to accommodate info every day; accordingly, now we have incorporated loads of introductory and educational fabric. any individual with the an identical of the maths history required for the graduate­ point examine of physics may be in a position to stick with the cloth contained during this publication, even though no longer with out eIfort. From the time the dissertation was once written formerly (approximately 365 days) our figuring out of the parameter estimation challenge has replaced largely. we have now attempted to include what we have now realized into this booklet. i'm indebted to a couple of those who have aided me in getting ready this docu­ ment: Dr. C. Ray Smith, Steve Finney, Juana Sunchez, Matthew Self, and Dr. Pat Gibbons who acted as readers and editors. moreover, i need to expand my private because of Dr. Joseph Ackerman for his aid through the time this manuscript was once being ready.

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Gjh i; D; I ) / exp 2h i jk This approximation will turn out to be very useful. g parameters as nuisances. For example, when we plot the power spectral density for multiple harmonic frequencies, we do not wish to plot this as a function of multiple variables, but as a function of one frequency: all other frequencies must be removed by integration. 16). There are two possible problems with this de nition of the power spectral density. First we assumed there is only one maximum in the posterior probability density, and second we asked a question about the total power carried by the signal, not a question about one spectral line.

But a glance at the data shows clearly that there is more present than our model assumed: therefore, probability theory must estimate the noise to be quite large. This suggests that we might do better by using a more realistic model which allows the \signal" to have more structure. Such a model can be t to the data more accurately; therefore it will estimate the noise to be smaller. This should permit a still better period estimate! But caution forces itself upon us; by adding more and more components to the model we can always t the data more and more accurately; it is absurd to suppose that by mere proliferation of a model we can extract arbitrarily accurate estimates of a parameter.

16). There are two possible problems with this de nition of the power spectral density. First we assumed there is only one maximum in the posterior probability density, and second we asked a question about the total power carried by the signal, not a question about one spectral line. It will turn out that the multiple frequency model will be invariant under permutations of the labels on the frequencies. It cannot matter 2 2 2 2 2 2 2 2 2 2 2 2 2 2 =1 2 The Power Spectral Density 53 which frequency is number one and which is labeled number two.

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