XG Firewall. Intercept X. For Home Users. Free Security Tools. Free Trials. Product Demos. Award-winning computer security news. Free tools Sophos Home for Windows and Mac. Hitman Pro. Sophos Mobile Security for Android. Virus Removal Tool. Antivirus for Linux. Other editions. Enlarge cover. Error rating book. Refresh and try again. Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. In this book Griffin responds to critiques of his earlier work--God, Power, and Evil: A Process Theodicy--and discusses ways in which his position has changed in the intervening years.
In so doing, he examines the problem of evil, theodicy, and philosophical theology, and contrasts traditional theism and process theism with regard to the question of omnipotence. Get A Copy. Hardcover , pages. More Details The maximum-likelihood improvement, H f , and best-fitting parameters for the planarity model i.
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However, different evidence measures reach different conclusions. All the measures find at least substantial evidence for the planarity model; however, the AIC and BIC appear to significantly overestimate this evidence compared to the BE result. We refer to Section 3. However, a totally new perspective into these instabilities now makes itself known.
H f only becomes the real evidence H after it is degraded by the penalization H p , related to the number of parameters of the model. This immediately removes the instabilities found in the frequentist formalism, by effectively penalizing for jumping between close calls, when one choice leads to a better common set of parameters. The penalization forces the multipoles to choose common parameters, at the risk of decreasing the fit a little.
The results are presented in Table 5. Thus as far as choice of statistics versus available data sets is concerned we have found an improved formalism and a robust set of best-fitting parameter values.
Regrettably at this point we see that the options for penalization spoil the party, with the BE and BIC finding no evidence for the m -preference model except for TOH1 , while the AIC favours the m -preference model over the base model, and the planarity model except for TOH3. We should perhaps not be overly disheartened by all this discord. We note that the BIC gives us a simple tool to examine the effect of priors.
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However, such a prior should be physically motivated. To assess in a frequentist way the significance of the maximum-likelihood values, H f , in Tables 2 and 5 we compare our results to those from simulations. We stress that this is an alternative to the Bayesian method, for which the evidence is completely summarized by the BE, with significance determined by the Jeffrey's scale.
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We plot histograms of the results in Fig. This approach provides us with an alternative measure of the significance of our H f values, and in Fig. We see that the planarity model consistently finds significance at the 98 per cent level. Note that it is this map that finds the m -preference AoE with the original statistic see Table 1.
The distribution of H f returned by 10 Gaussian and isotropic simulations for the planarity model left-hand panel and the general m -preference model middle panel. We also plot the result obtained by the WMAP3 map short-dashed line. In the table, we list the percentage of simulations that find higher H x f values for the planarity model P and the m -preference model m.
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We stress that this approach does not take account of the relative complexities of the models. However, the Bayesian approach generally finds lower evidence for these models compared to the base model, and it actually finds no evidence for the m -preference model except for TOH3.
This reflects the well-known fact that the Bayesian approach to model selection tends to set a higher threshold than frequentist approaches e. Trotta ; Mukherjee et al.
A disadvantage of the Bayesian approach is its sensitivity to priors, and its insensitivity to useless parameters that are unconstrained by the data. However, the frequentist approach can involve a large amount of computational time and can be prone to selection effects. Consider that we could always choose some convoluted complex statistic for which our data return anomalously high or low values, compared to the simulations. Only the Bayesian approach can help here in imposing a suitable penalization, by averaging the likelihood over the extra parameter space.
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This ensures that a model is preferred only if the improvement in the fit merits opening up this extra dimension of parameter space. The IC method provides another way of penalizing for the extra parameters; however, we see that the AIC generally prefers the m -preference model with the most parameters to the planarity or base model — in disagreement with both the BE and the frequentist approach. These are primarily: i lack of robustness: small changes in the data produce very different best-fitting parameter values, that is, the statistics are discontinuous; ii variations with data set: it is hard to connect varying results to imperfections in the data or the statistic; and iii the need for simulations to assess significance: no way of penalizing for extra parameters or comparing competing theories on an equal footing, for example, planarity versus general m-preference.
We have found an improved formalism by employing a model selection approach, which cures the instabilities by favouring common parameters between the multipoles. The original instabilities were due to the existence of multiple solutions for a given multipole. We now find that the best-fitting parameter values are robust.
Using the BE and the BIC approximations, we find that there is substantial evidence for the planarity model, but no evidence for the m -preference model. These results are in contradiction with the AIC approach which finds evidence for both models, and generally stronger evidence for the m -preference model. We think that this demonstrates a weakness of this crude statistic, that does not appear to penalize enough for extra parameters. The m -preference model is a more general version of the planarity model.