tokens: priors, the relative magnitudes of each component are able to vary freely. While fits performed without priors can produce biased results if the training sets cannot sufficiently describe the shapes of the components, the use of a prior derived from the training set applies constraints on both the shapes and the magnitudes of the components. Thus, caution should be taken to ensure that there is a high level of confidence that the given training set is sufficient to describe both the shape and magnitude of the component before a training set prior is applied. 12 12 In principle, if one is uncertain that the training set is fully representative of the data, it may be possible to apply a “weak” prior, which falls in between the highly constrained and uniform priors used in the examples here. A complicating factor in applying this method is that the posterior will no longer be Gaussian if the prior takes a non-Gaussian form. Note that when using a prior distribution, instead of minimizing the DIC in order to choose the number of SVD modes, we instead maximize the Bayesian evidence , given by where the parameters x are the coefficients of the SVD modes, is the likelihood, is the model composed of the SVD modes, and π ( x ) is the prior distribution. Note that the dimensionality N p of the parameter space will change when the number of SVD modes to be fit is varied. Maximizing the evidence ensures that the prior is properly taken into account. When using a training set prior, variations in modes unimportant to the training set are suppressed, but a larger number of modes may still be chosen by the evidence. Thus, the number of SVD modes may no longer provide as much information about the quality of the estimate of the signal. However, if the training sets do not encompass the true form of one or more of the components, the constraints provided by the prior will prevent the full data from being fit well. In this case, goodness-of-fit statistics such as or will be able to identify a poor fit. Further details regarding the use of these metrics for fits with priors are given in Section 3 . 2.4. Test Strategy In this section, we present a step-by-step outline for assessing the quality of the signal extraction in an observational setting when the true form of the signal is unknown. This test can be performed for fits with or without priors; however, we suggest that the test be performed without a prior distribution first owing to the fact that the use of an incorrect prior distribution will lead to significant bias in the extracted signal, as described in Section 2.3 . If the fit “passes” the test as described below, one may wish to include a training set prior to possibly improve the quality of the signal reconstruction further. 1. Fit many data realizations created from the training set curves. If no prior is used, create a distribution of the chosen numbers of SVD modes. This distribution will have the same dimensionality as the number of
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O B-Model O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O