J. Skilling, D.R.T. Robinson, and S.F. Gull. Probabilistic displays. In Jr. W.T. Grandy and L.H. Schick, editors, Maximum Entropy and Bayesian Methods, pages 365--368. Kluwer Academic, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....posterior of the models used to interpret data. Part of the reason reliability has not been approached by many lies in the di#culty of doing so, particularly in a large dimensional space. We know of two ways to visualize the reliability of inferred models. The first, proposed by Skilling et al. [22], provides a stochastic look at the range of possible solutions. It involves the display of a sequence of solutions that are randomly chosen from the posterior probability distribution. This sequence, typically calculated o# line, is presented as a video loop. By showing a representative range of ....

....and variance, which is one way to summarize the uncertainty in solutions. This technique permits marginalization with respect to any parameters that are not of interest. It also can provide a visualization of the uncertainty in solutions by displaying as a video loop the sequence of random samples [22]. Most of the models presently available in the BIE are two dimensional. We expect to develop 1D models soon. One dimensional models will provide a very useful environment in which to develop demonstrations of how the Bayesian approach addresses many familiar problems such as deconvolution, or ....

J. Skilling, D. R. T. Robinson, and S. F. Gull, "Probabilistic displays," in Maximum Entropy and Bayesian Methods, W. T. Grandy, Jr. and L. H. Shick, eds., pp. 365--368, Kluwer Academic, Dordrecht, 1991.

....the minus log prior. This technique seems to be an essential tool to deal with the issue of uncertainties in estimated parameters. One valuable use of MCMC is to visualize the degree of model variation allowed by the uncertainties, which only requires that one has a way of displaying the model [29, 23]. Most useful is the ability to characterize the uncertainty in the model in whatever way one wishes, e.g. in terms of the variance in the parameters or of quantities derived from the model. In uncertainty assessment, it is crucially important to include the effects of correlations among the ....

J. Skilling, D. R. T. Robinson, and S. F. Gull, "Probabilistic displays," in Maximum Entropy and Bayesian Methods, W. T. Grandy, Jr. and L. H. Shick, eds., pp. 365-- 368, Kluwer Academic, Dordrecht, 1991.

....analysis been developed and applied to real world problems. This paper will review Bayesian model comparison, regularisation, and noise estimation, by studying the problem of interpolating noisy data. The Bayesian framework I will describe for these tasks is due to Gull and Skilling [5, 6, 8, 17, 18], who have used Bayesian methods to achieve the state of the art in image reconstruction. The same approach to regularisation has also been developed in part by Szeliski [22] Bayesian model comparison is also discussed by Bretthorst [2] who has used Bayesian methods to push back the limits of ....

....to the full covariance information for the entire interpolant, not just the pointwise error bars. It is possible to visualise the joint error bars on the interpolant by making typical samples from the posterior distribution, performing a random walk around the posterior bubble in parameter space [18]. It is simple to program such a random walk for interpolation problems such as are examined in this paper; however for lack of dynamic paper I will have to leave a demonstration of this to your imagination. In this section objective comparison of alternative models will be demonstrated; this ....

J. Skilling, D.R.T. Robinson, and S.F. Gull (1991). Probabilistic displays, in [4], 365--368.

....lower than its optimal value. Because successive samples in an MCMC sequence are highly correlated, these five samples are separated by 2000 steps to minimize correlations between them. This kind of display of model realizations is a good way to visualize the characteristics of an inferred model [13] 8 KENNETH M. HANSON AND JANE M. BOOKER (a) b) Figure4. Five widely separated samples from an MCMC sequence, shown in both (a) the data domain and (b) the alpha domain. The hyperparameter is 0.04 for this case. Viewing these samples provides one with an understanding of the type of curves that ....

J. Skilling, D. R. T. Robinson, and S. F. Gull, "Probabilistic displays," in Maximum Entropy and Bayesian Methods, W. T. Grandy, Jr. and L. H. Shick, eds., pp. 365--368, Kluwer Academic, Dordrecht, 1991.

....lead to substantial calculational inefficiency. This inefficiency results in correlation between successive samples taken from the MCMC sequence, which means that a reduced number of samples can adequately represent the full sequence. A number of schemes for improving the efficiency of MCMC exist [26 30], most of which are adaptive, and many of which require the gradient of with respect to the parameters. Therefore, adjoint differentiation can be very helpful in making MCMC more efficient. 3 Uncertainty in simulation code predictions The goal of the present work is to determine the ....

J. Skilling, D. R. T. Robinson, and S. F. Gull, Probabilistic displays, in Maximum Entropy and Bayesian Methods, edited by W. T. Grandy, Jr. and L. H. Shick (Kluwer Academic, Dordrecht, 1991) 365.

....and applied to more complex problems in other fields. This paper will review Bayesian model comparison, regularisation, and noise estimation, by studying the problem of interpolating noisy data. The Bayesian framework I will describe for these tasks is due to Gull (1988, 1989a, 1991) and Skilling (1991), who have used Bayesian methods to achieve the state of the art in image reconstruction. The same approach to regularisation has also been developed in part by Szeliski (1989) Bayesian model comparison is also discussed by Bretthorst (1990) who has used Bayesian methods to push back the limits ....

....information for the entire interpolant, not just the pointwise error bars. It is possible to visualise the joint error bars on the interpolant by making typical samples from the posterior distribution, performing a random walk around the posterior bubble in parameter space (Sibisi, 1991, Skilling et al. 1991). Figure 8 shows data set Y interpolated by three typical interpolants found by random sampling from the posterior distribution. These error bar properties are found under the assumption that the model is correct; so it is possible for the true interpolant to lie significantly outside the error ....

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J. Skilling, D.R.T. Robinson, and S.F. Gull (1991). `Probabilistic displays', in Grandy and Schick, eds. (1991), 365--368.

....in Fig. 12, and correspond to nalL = 1:51. The actual error bars submitted were scaled to much smaller values due to a missinterpretation of the performance measure, nalL = 3:5. Alternative Bayesian methods for estimating uncertainties have been suggested by MacKay [35, 36] and Skilling [37]. Additional predictions: The complete 10000 point series continuation was provided after the competition. Fig. 13 shows various iterated predictions starting at different locations within 0 50 100 150 200 250 1000 1010 1020 1030 1040 1050 1060 1070 1080 1090 1100 Figure 12: Estimated standard ....

J. Skilling, D. Robinson, S. Gull, "Probabilistic displays", in Maximum Entropy and Bayesian Methods, Laramie, 1990, W Grandy and L. Schick, eds., pp.365-368. Kluwer, Dordrecht.

....the uncertainties in any two parameters, it does not provide much insight. One appealing way to get a feeling for the uncertainty in a Bayesian solution is to display a sequence of distinct solutions drawn from the posterior probability distribution. This approach was suggested by Skilling et al. [2], who produced a video display of a random walk through the posterior distribution. However, the calculational method used in that work was based on a Gaussian approximation of the posterior probability distribution in the neighborhood of the MAP solution. Later Skilling made some progress in ....

J. Skilling, D.R.T. Robinson, and S.F. Gull. Probabilistic displays. In Jr. W.T. Grandy and L.H. Schick, editors, Maximum Entropy and Bayesian Methods, pages 365-368. Kluwer Academic, 1991.

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J. Skilling, D. R. T. Robinson, and S. F. Gull. Probabilistic displays. In Grandy and Schick

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J. Skilling, D.R.T. Robinson, and S.F. Gull. Probabilistic displays. In Jr. W.T. Grandy and L.H. Schick, editors, Maximum Entropy and Bayesian Methods, pages 365--368. Kluwer Academic, 1991.

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J. Skilling, D.R.T. Robinson, and S.F. Gull (1991). Probabilistic displays, in [4], 365--368.

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