Likelihood maximization

A parametric method for handling missing data is maximum likelihood. Recall that in linear regression maximum likelihood maximizes the likelihood function L(.)... 

Since we got the testing data, we usually construct the likelihood function i( x) of the model parameters B, and maximize the likelihood directly by numerical methods, such as Newton-Raphson method, to obtain the estimators 0. However, in the case of competing failure modes, if the number of causes is large, L(0 x) can become overparameterized, and the direct likelihood maximization often leads to rmstable numerical results (Bousquet, Bertholon and Celeux 2006). The difficulty can be overcome through the use of the EM algorithm for MLE. 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

Motivation Unit tests require a substantial investment in time and resources to complete successfully. This is the case whether the test is a straightforward analysis of pump performance or a complex analysis of an integrated reactor and separation train. The uncertainties in the measurements, the likelihood that different underlying problems lead to the same symptoms, and the multiple interpretations of unit performance are barriers against accurate understanding of the unit operation. The goal of any unit test should be to maximize the success (i.e., to describe accurately unit performance) while minimizing the resources necessary to arrive at the description and the subsequent recommendations. The number of measurements and the number of trials should be selected so that they are minimized. 

Truncating the plane constrains the centroid estimate to a certain region, making the variance finite. Since the truncated plane is placed where the centre is expected to be we are implicitly adding prior information (van Dam and Lane, 2000). The smaller the plane, the more the centroid is effectively localized and the more prior information is assumed. Therefore, by adding prior information, truncating the plane can improve the centroid estimate, even though some photons are lost. The optimal solution is to maximize the likelihood directly. 

The maximum likelihood (ML) solution is the one which maximizes the probability of the data y given the model among all possible x ... 

The optimize command maximizes a statistical "likelihood function". The higher this function, the more likely is the parameter to be the correct one. In the figure below, the symbols represent points calculated by the program Topaz (the full model), and the solid lines are the values calculated from the reduced-order model using the parameters determined by the program. 

This likelihood function has to be maximized for the parameters in f. The maximization is to be done under a set of constraints. An important constraint is the knowledge of the peak-shapes. We assume that f is composed of many individual... 

If the covariance matrices of the response variables are unknown, the maximum likelihood parameter estimates are obtained by maximizing the Loglikeli-hood function (Equation 2.20) over k and the unknown variances. Following the distributional assumptions of Box and Draper (1965), i.e., assuming that i= 2=...=En= , it can be shown that the ML parameter estimates can be obtained by minimizing the determinant (Bard, 1974)... 

Since POAG is a chronic, often asymptomatic condition, the decision of when and how to treat patients is difficult since the treatment modalities are often expensive and have potential adverse effects or complications. The clinician should evaluate the potential effectiveness, toxicity, and the likelihood of patient adherence for each therapeutic modality. The ideal therapeutic regimen should have maximal effectiveness and patient tolerance to achieve the desired therapeutic response. The American Academy of Ophthalmology (AAO) publishes Preferred Practice Patterns for POAG and POAG Suspect.2... 

Bricogne, G. (1993) Direct phase determination by entropy maximization and likelihood ranking status report and perspectives, Acta Cryst., D49, 37-60. 

We have observed 3He towards several PNe that have been selected to maximize the likelihood of 3He+ detections. First epoch observations with the GBT are discussed in [3]. Figure 2 shows a 4 a detection towards the PNe J 320 with the VLA. Both of these results are consistent with 3He/H abundances between 10 4 — 10 3 by number and standard stellar evolution models. Observations with Arecibo are planned for winter 2005. Our goal is to be able to make a connection between some of the selection criteria and a high 3He abundance. In this way we can use subsidiary measures, e.g., the N abundance, to estimate what fraction of PNe have preserved their 3He. 

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. 

Maximum likelihood method The estimate of a parameter 9, based on a random sample Xi, X2, , Xn, is that value of 9 which maximizes the likelihood function L(Xi, X2, , Xn, 9) which is defined as... 

With regard to the design of the test, mice are mated when 7-8 weeks old. By this age all germ cell stages are present. The test compound is normally administered by the IP route to maximize the likelihood of germ cell exposure. The preferred dose is just below the toxic level so long as fertility is not compromised. One lower dose should also be included. 

The principle of maximum likelihood tells us that we should use as our estimate that value which maximizes the likelihood of the observed event. 

The commercially available software (Maximum Entropy Data Consultant Ltd, Cambridge, UK) allows reconstruction of the distribution a.(z) (or f(z)) which has the maximal entropy S subject to the constraint of the chi-squared value. The quantified version of this software has a full Bayesian approach and includes a precise statement of the accuracy of quantities of interest, i.e. position, surface and broadness of peaks in the distribution. The distributions are recovered by using an automatic stopping criterion for successive iterates, which is based on a Gaussian approximation of the likelihood. 

This algorithm is well-known under the name expectation maximization algorithm (EM) (McLachlan and Pee 2000). Since the parameters /xr Xj, and pj are already needed in the E-step for computing the likelihoods, these parameters have to be... 

The solution for model-based clustering is based on the Expectation Maximi-zation (EM) algorithm. It uses the likelihood function and iterates between the expectation step (where the group memberships are estimated) and the maximization step (where the parameters are estimated). As a result, each object receives a membership to each cluster like in fuzzy clustering. The overall cluster result can be evaluated by the value of the negative likelihood function which should be as small as possible. This allows judging which model for the clusters is best suited (spherical clusters, elliptical clusters) and which number of clusters, k, is most appropriate. 

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