Likelihood function

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. 

Setting up the probability model for the data and parameters of the system under smdy. This entails defining prior distributions for all relevant parameters and a likelihood function for the data given the parameters. 

We need a mathematical representation of our prior knowledge and a likelihood function to establish a model for any system to be analyzed. The calculation of the posterior distribution can be perfonned analytically in some cases or by simulation, which I... 

Any data set that consists of discrete classification into outcomes or descriptors is treated with a binomial (two outcomes) or multinomial (tliree or more outcomes) likelihood function. For example, if we have y successes from n experiments, e.g., y heads from n tosses of a coin or y green balls from a barrel filled with red and green balls in unknown proportions, the likelihood function is a binomial distribution ... 

In practice, it may not be possible to use conjugate prior and likelihood functions that result in analytical posterior distributions, or the distributions may be so complicated that the posterior cannot be calculated as a function of the entire parameter space. In either case, statistical inference can proceed only if random values of the parameters can be drawn from the full posterior distribution ... 

The likelihood function is an expression for p(a t, n, C), which is the probability of the sequence a (of length n) given a particular alignment t to a fold C. The expression for the likelihood is where most tlireading algorithms differ from one another. Since this probability can be expressed in terms of a pseudo free energy, p(a t, n, C) x exp[—/(a, t, C)], any energy function that satisfies this equation can be used in the Bayesian analysis described above. The normalization constant required is akin to a partition function, such that... 

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. 

Ideally, to characterize the spatial distribution of pollution, one would like to know at each location x within the site the probability distribution of the unknown concentration p(x). These distributions need to be conditional to the surrounding available information in terms of density, data configuration, and data values. Most traditional estimation techniques, including ordinary kriging, do not provide such probability distributions or "likelihood of the unknown values pC c). Utilization of these likelihood functions towards assessment of the spatial distribution of pollutants is presented first then a non-parametric method for deriving these likelihood functions is proposed. 

This conditional pdf can be seen as the likelihood function of the unknown value p(x) ... 

An answer to the previous problems is provided by the conditional distribution approach, whereby at each node x of a grid the whole likelihood function of the unknown value p(x) is produced instead of a single estimated value p (x). This likelihood function allows derivation of different estimates corresponding to different estimation criteria (loss functions), and provides data values-dependent confidence intervals. Also this likelihood function can be used to assess the risks a and p associated with the decisions to clean or not. 

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... 

Algebraic equations Steady state of CSTR with first-order kinetics. Algebraic solution and optimisation (least squares. Draper and Smith, 1981). Steady state of CSTR with complex kinetics. Numerical solution and optimisation (least squares or likelihood function). 

Under the simplifying assumption that the reflexions are independent of each other, K, can be written as a product over reflexions for which experimental structure factor amplitudes are available. For each of the reflexions, the likelihood gain takes different functional forms, depending on the centric or acentric character, and on the assumptions made for the phase probability distribution used in integrating over the phase circle for a discussion of the crystallographic likelihood functions we refer the reader to the description recently appeared in [51]. 

Both the a priori and the likelihood functions contain exponentials, so that it is convenient to consider the logarithm of the posterior probability, and maximise the Bayesian score ... 

In this section we briefly discuss an approximate formalism that allows incorporation of the experimental error variances in the constrained maximisation of the Bayesian score. The problem addressed here is the derivation of a likelihood function that not only gives the distribution of a structure factor amplitude as computed from the current structural model, but also takes into account the variance due to the experimental error. 

We have for now implemented a drastic simplification, whereby the likelihood function is taken equal to the error-free likelihood, but to the variance parameter Z2 appearing in the latter function the experimental error variance is added ... 

This approximation has already proven very effective in the calculation of likelihood functions for maximum likelihood refinement of parameters of the heavy-atom model, when phasing macromolecular structure factor amplitudes with the computer program SHARP [53]. A similar approach was also used in computing the variances to be used in evaluation of a %2 criterion in [54]. 

BUSTER has been run against the L-alanine noisy data the structure factor phases and amplitudes for this acentric structure were no longer fitted exactly but only within the limits imposed by the noise. As in the calculations against noise-free data, a fragment of atomic core monopoles was used the non-uniform prior prejudice was obtained from a superposition of spherical valence monopoles. For each reflexion, the likelihood function was non-zero for a set of structure factor values around this procrystal structure factor the latter acted therefore as a soft target for the MaxEnt structure factor amplitude and phase. 

Goodness-of-fit tests may be a simple calculation of the sum of squared residuals for each organ in the model [26] or calculation of a log likelihood function [60], In the former case,... 

SSR = sum of squared residuals N = number of observations C° = mean of the observed drug concentration C = predicted drug concentration rii = number of experimental repetitions S2 = variance of the observed concentrations at each data point LL = log likelihood function... 

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... 

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

The parameter estimation for the mixture model (Equation 5.25) is based on maximum likelihood estimation. The likelihood function L is defined as the product of the densities for the objects, i.e.,... 

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. 

By interpreting the state vector u in y (x t) and the position vector x as a parameter, In y (x, f) can be interpreted as a likelihood function and / (x, t) as a vector of statistical scores as expected from likelihood theory, the average score is zero, Xu( > 0 = Pq. (4)]. Moreover, the covariance matrix of the relative rates of evolution plays the role of a Fisher information metric ... 

The criterion for best fit is based on the maximum likelihood principle (Fisher 1922) where the best estimates of the model parameters should maximise the likelihood function, L, for the observation of N different experimental observations. 

ML estimation optimizes the likelihood function. Use the optimized value of the log-likelihood function. 

ML is the approach most commonly used to fit a distribution of a given type (Madgett 1998 Vose 2000). An advantage of ML estimation is that it is part of a broad statistical framework of likelihood-based statistical methodology, which provides statistical hypothesis tests (likelihood-ratio tests) and confidence intervals (Wald and profile likelihood intervals) as well as point estimates (Meeker and Escobar 1995). MLEs are invariant under parameter transformations (the MLE for some 1-to-l function of a parameter is obtained by applying the function to the untransformed parameter). In most situations of interest to risk assessors, MLEs are consistent and sufficient (a distribution for which sufficient statistics fewer than n do not exist, MLEs or otherwise, is the Weibull distribution, which is not an exponential family). When MLEs are biased, the bias ordinarily disappears asymptotically (as data accumulate). ML may or may not require numerical optimization skills (for optimization of the likelihood function), depending on the distributional model. 

Implementation of ML is straightforward in many cases. More difficult situations may involve a need to incorporate random effects, covariates, or autocorrelation. The likelihood function may involve difficult or intractable integrals. However, recent developments in statistical computing such as the EM algorithm and Gibbs sampler provide substantial flexibility for such cases (in complicated situations, a specialist in current statistical computing may be helpful). Alternatively, the GLS approach described below may be applicable. 

In this example, the likelihood function is the distribution on the average of a random sample of log-transformed tissue residue concentrations. One could assume that this likelihood function is normal, with standard deviation equal to the standard deviation of the log-transformed concentrations divided by the square root of the sample size. The likelihood function assumes that a given average log-tissue residue prediction is the true site-specific mean. The mathematical form of this likelihood function is... 

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