Threshold maximum likelihood

The geometric mean of the individual thresholds (maximum likelihood threshold). 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

A few court decisions, however, have been more skeptical of the linear model. Eor example, the U.S. EPA s use of the linear, no-threshold model in its risk assessment for drinking water chlorinated byproducts was rejected by the court because it was contrary to evidence suggesting a nonlinear model that had been accepted by both the U.S. EPA and its Science Advisory Board (CCC 2000). On the other hand, the U.S. OSHA s departure from the linear, no-threshold model in its formaldehyde risk assessment was likewise rejected by the court (lU 1989). The court held that the U.S. OSHAhad improperly used the maximum likelihood estimate (MLE) rather than the upper confidence limit (UCL) to calculate risk, and the UCL but not the MLE model was consistent with a linear dose-response assumption. The court held that the U.S. OSHA had failed to justify its departure from its traditional linear, no-threshold dose-response assumption. 

Fig. 11 Data rectification of signal with deterministic features. Dashed line is noisy data, (a) Original and noisy data, (h) Wavelet thresholding, (c) Wavelet thresholding after maximum likelihood rectification, (d) Multi.scale Bayesian rectification.

MR Linschoten, LO Harvey, Jr, PM Eller, BW Jafek. Fast and accurate measurement of taste and smell thresholds using a maximum-likelihood adaptive staircase procedure. Percept Psychophys 63 1330-1347, 2001. 

Using the field model described in section 1, detection probabilities are to be computed for each grid point to find the breach probability. The optimal decision rule that maximizes the detection probability subject to a maximum allowable false alarm rate a is given by the Neyman-Pearson formulation [20]. Two hypotheses that represent the presence and absence of a target are set up. The Neyman-Pearson (NP) detector computes the likelihood ratio of the respective probability density functions, and compares it against a threshold which is designed such that a specified false alarm constraint is satisfied. 

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