Penalized likelihood

Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its properties. Journal of the American Statistical Association, 96, 1348-1360. 

Sanderson, M.J., Estimating absolute rates of molecular evolution and divergence times a penalized likelihood approach, Mol. Biol. EvoL, 19, 101, 2002. 

Rate Estimates Derived from An rSs Analysis Using Penalized Likelihood, Globally Constant and Non-Parametric Forms of Rate-Smoothing... 

Crown Clade Group Penalized Likelihood Globally Constant Non-P rametric... 

Accordingly, a penalized likelihood version of t-statistic has been proposed and implemented. For example, Wu (2005) proposed another modified t-statistics, taking advantage of both SAM and lasso methods. The method works as follows assume the linear regression situation. 

Lanteri, H., Roche, M., and Aime, C., 2002, Penalized maximum likelihood image restoration with positivity constraints multiplicative algorithms. Inverse Problems 18, 1397... 

The score/penalty assigned to a match/mismatch is dependent on the likelihood that this match/mismatch occurred by chance alone. If an event occurs randomly with a high frequency the score will be small, while very rare events will receive very large scores. For example, a conservative amino-acid substitution will receive a very small penalty, while the introduction of a stop codon will be penalized heavily. A variety of scoring schemes have been generated based on factors such as the bio-physical character, evolutionary distance, and the subcellular localization of the sequences being compared. 

A.3.6 Maximum Penalized Marginal Likelihood (MPML) Approach... 

To summarize, in the Bayesian approach to model selection, the model classes are ranked according to p V Cj)P(Cj U) for 7 = 1,2,..., Nc, where the most plausible class of models representing the system is the one which gives the largest value of this quantity. The evidence p V Cj) can be calculated for each class of models using Equation (6.11) where the likelihood p V 9, Cj) is evaluated using the methods presented in Chapters 2-5. The prior distribution P Cj U) over all the model classes Cj, j = 1,2,..., Nc, can be used for other concerns, such as computational demand. However, it is out of the scope of this book and uniform prior plausibilities are chosen, leaving the Ockham factor alone to penalize the model classes. 

It is clear that the prior distribution affects the modal class selection results. Therefore, the choice of prior distribution is important for model class selection because it offers a reference for comparison in quantifying the information gained from the data. The prior distribution expresses how much previous experience or information a user has about a model class. A more informative prior distribution is used if the user has more experience of the model class. The evidence of such a model class is surplus due to the lifting of the prior PDF. However, inappropriate previous information on the parameters will be penalized by the small value of the inner product of the prior distribution and the likelihood function. In general, it is more difficult to give the prior distribution for empirical models since the physical meaning of the parameters are not as obvious as physical models. More investigations are needed to explore further in this direction. 

The Act established three violation categories (1) nonserious, (2) serious, and (3) willful and repeated. The law does not require OSHA inspectors to penalize companies for nonserious violations of health and safety standards. Inspectors must, however, levy fines for serious violations, which are infractions substantially increasing the likelihood of a worker fatality or serious injury, and to levy fines for willful and repeated violations. Under the original Act inspectors could fine firms up to 1,000 for each serious violation of health and safety standards, 10,000 for each willful violation, and 1,000 per day for each repeated violation. In 1990 Congress increased maximum fines to 7,000 for serious violations and increased maximum fines to 70,000 for willful and repeated violations. The Occupational Safety and Health Act established criminal penalties only in situations where willful violations result in worker deaths. ... 

Moons, K.G.M., Donders, A.R.T., Steyerberg, E.W. Harrell, FE. 2004 Penalized maximum likelihood estimation to directly adjust diagnostic and prognostic prediction models for overoptimism a clinical example. Journal of Clinical Epidemiology, 57, 1262-1270. 

Among our leading industrial clients, the companies setting the benchmark for industry safety often have the highest rates of reported near misses because they do not penalize the reporting of near misses and do not directly reward the reduction of incident rates. Instead, they welcome the information stemming from near misses, quickly digest its implications, and act immediately to reduce the likelihood of repeated exposures to hazard. 

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