Jensen-Shannon Divergence as a Goodness-of-Fit Measure for Maximum Likelihood Estimation and Curve Fitting
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The coefficient of determination, known as R^2, is commonly used as a goodness-of-fit criterion for fitting linear models. R^2 is somewhat controversial when fitting nonlinear models, although it may be generalised on a case-by-case basis to deal with specific models such as the logistic model. Assume we are fitting a parametric distribution to a data set using, say, the maximum likelihood estimation method. A general approach to measure the goodness-of-fit of the fitted parameters, which we advocate herein, is to use a nonparametric measure for model comparison between the raw data and the fitted model. In particular, for this purpose we put forward the Jensen-Shannon divergence (JSD) as a metric, which is bounded and has an intuitive information-theoretic interpretation. We demonstrate, via a straightforward procedure making use of the JSD, that it can be used as part of maximum likelihood estimation or curve fitting as a measure of goodness-of-fit, including the construction of a confidence interval for the fitted parametric distribution. We also propose that the JSD can be used more generally in nonparametric hypothesis testing for model selection.
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