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Topic: Likelihood function

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	Likelihood function - Wikipedia, the free encyclopedia
		Likelihood as a solitary term is a shorthand for likelihood function.
		That is, the likelihood function for B is the equivalence class of functions
		The likelihood function is not a probability density function – for example, the integral of a likelihood function is not in general 1.
	en.wikipedia.org /wiki/Likelihood_function (823 words)

	PlanetMath: score function
		The likelihood equations may also be formed by setting the gradient of the plain likelihood function to zero.
		The use of the log function often facilitates the algebra as many distributions are exponential in nature.
		This is version 9 of score function, born on 2004-07-08, modified 2006-09-23.
	planetmath.org /encyclopedia/LikelihoodEquation.html (276 words)

	PlanetMath: likelihood function
		In other words, the likelikhood function is functionally the same in form as a probability density function.
		Many of the density functions are exponential in nature, it is therefore easier to compute the MLE of a likelihood function
		This is version 10 of likelihood function, born on 2004-07-08, modified 2006-09-23.
	planetmath.org /encyclopedia/LikelihoodFunction.html (204 words)

	Tools - Estimation Methods - Marginal Maximum Likelihood - Details
		Multiplying these probabilities together for all possible proficiency levels is the basis of the likelihood function for the marginal maximum likelihood estimate.
		The overall likelihood function is the product of all of the individual likelihood functions for individuals in the sample.
		estimates as the likelihood function reaches its maximum (that is, an change in the likelihood always corresponds to a change in the same direction of the log-likelihood); and 2) logs can be summed over the observations, and sums are easier to work with than products.
	am.air.org /help/NAEPTextbook/htm/dmarginalmaximumlikelihood.htm (493 words)

	Maximum Likelihood
		Often, it is useful to think of a function whose domain is the sample space, W. Such a function is known as a random variable.
		Here, the likelihood calculated under the null hypothesis is in the numerator and the likelihood calculated under the alternative hypothesis is in the denominator.
		Likelihood: The likelihood function is used to obtain maximum likelihood estimates (MLEs) of parameters and is the probabiity of the observed sequences as a function
	ucjeps.berkeley.edu /bryolab/ib200/maximumlikelihood.html (2941 words)

	The Likelihood Function (Site not responding. Last check: 2007-11-03)
		This function looks almost identical to the probability function, but notice a subtle difference: in the probability function the parameter p is assumed to be fixed, and the observation (x) is assumed to be variable (and a function of p and n).
		In the likelihood we already have the data, and it’s p we are trying to determine; p is now a function of the data.
		The last window, which represents the derivative of this function with respect to p, shows the function that is actually solved to obtain the MLE (by setting it equal to zero and solving for p).
	fisher.forestry.uga.edu /popdyn/Likelihood.html (669 words)

	Parameter Estimation Techniques (Site not responding. Last check: 2007-11-03)
		The likelihood function for a given distribution is a representation of the probability of obtaining the sample data.
		For the assumed distribution, the cumulative distribution function is transformed to a linear expression, usually by a logarithmic transformation, and plotted.
		Maximum likelihood estimation provides no indication of goodness of fit, thus it is recommended to verify the fit of the chosen distribution with either a hazard plot or a probability plot, then use maximum likelihood estimation for all inferences concerning the population.
	www.engineeredsoftware.com /lmar/pe_techniques.htm (600 words)

	Molecular Replacement
		In the direct rotation function, the molecule is placed in the unit cell of the unknown structure, and the Patterson for the oriented molecule is compared with the entire Patterson of the unknown structure.
		a correlation function of the density with itself), the intermolecular vector set is a correlation function between the density for one molecule and the density for the other molecule, as shown in the following figure and equation.
		For instance, when you are computing a traditional rotation function, you have to choose the resolution range of the data, whether or not to use normalised or sharpened data (and if so, sharpened by how much), integration radii, and whether or not to include side chains.
	www-structmed.cimr.cam.ac.uk /Course/MolRep/molrep.html (3684 words)

	[No title]
		For the Sigma(A) curve used to calibrate the likelihood function, there must be an estimate of the effective RMS error of the model.
		Refinement was conducted in CNX using a maximum likelihood function that includes the experimental phases as part of the target (33).
		For all the considered models the parameters were estimated by maximising the likelihood function.
	www.lycos.com /info/likelihood-function--models.html (519 words)

	Bayes's Theorem and the Likelihood Function
		What this means, roughly, is that the likelihood function looks like a bell curve with a peak at 1.02 and a width (standard deviation) of 0.02, roughly as shown in Figure 2.
		The likelihood function, by definition, is the probability density of getting the data that we actually observed, as a function of the value of
		(where the likelihood is extremely small), the data that we actually observed would have been quite unlikely to occur.
	www.richmond.edu /~ebunn/bayes/node4.html (597 words)

	Maximum Likelihood Estimation
		For the likelihood to be properly thought of as a density, a Bayesian approach is required.
		The iteration procedes until a local maximum of the likelihood is attained, although in the case of the first two methods, such convergence is not guaranteed.
		The EM algorithm has the advantage that the likelihood is always increased at each iteration, and so convergence to at least a local maximum is guaranteed (assuming a bounded likelihood).
	cnx.org /content/m11446/latest (1936 words)

	Maximum Likelihood Function (Site not responding. Last check: 2007-11-03)
		Maximum likelihood estimation endeavors to find the most "likely" values of distribution parameters for a set of data by maximizing the value of what is called the "likelihood function." This likelihood function is largely based on the probability density function (pdf) for a given distribution.
		Since the log-likelihood function is easier to manipulate mathematically, we derive this by taking the natural logarithm of the likelihood function.
		Thus, the "peak" of the likelihood surface function corresponds to the values of the parameters that maximize the likelihood function, i.e.
	www.weibull.com /hotwire/issue33/relbasics33.htm (1165 words)

	Apophenia: Apophenia: Maximum likelihood estimation
		Most of the action with regards to maximum likelihood estimation is in the function apop_maximum_likelihood and the model objects.
		The likelihood function is taken from the model, the metric is the Manhattan metric, the copy/destroy functions are just the usual vector-handling fns., et cetera.
		This functions is used in the sample code in the Maximum likelihood estimation section.
	apophenia.sourceforge.net /doc/group__mle.html (1165 words)

	Maximum likelihood - Wikipedia, the free encyclopedia
		It also assumes they are familiar with standard basic techniques of maximizing continuous real-valued functions, such as using differentiation to find a function's maxima.
		This result is easily generalized by substituting a letter such as t in the place of 49 to represent the observed number of 'successes' of our Bernoulli trials, and a letter such as n in the place of 80 to represent the number of Bernoulli trials.
		Since the logarithm is a continuous strictly increasing function over the range of the likelihood, the values which maximize the likelihood will also maximize its logarithm.
	en.wikipedia.org /wiki/Maximum_likelihood (1317 words)

	Maximum Likelihood Estimation (Site not responding. Last check: 2007-11-03)
		The point where the derivative of the likelihood function is zero and the second derivative is negative is the maximum.
		It turns out, that working with the log of the likelihood function is usually easier than working with the likelihood function itself.
		The results for the log-likelihood function hold for the likelihood function (the maximum is the same), so logging it does no damage.
	www.columbia.edu /~ag2319/teaching/G4075_Outline/node13.html (811 words)

	3 The Cox Proportional Hazards Model
		Each event time contributes one factor to the likelihood function; for tied events, all events in the tie appear with the same denominator.
		In the case of tied events, each of the events in the tie contributes its own term to the sum; this term is the same for all events in a particular tie.
		is monotonously increasing, and the survival function estimates are monotonously decreasing.
	sunsite.univie.ac.at /XploRe/tutorials/haznode4.html (1921 words)

	8.4.1.2. Maximum likelihood estimation
		Maximum likelihood estimation begins with writing a mathematical expression known as the Likelihood Function of the sample data.
		Loosely speaking, the likelihood of a set of data is the probability of obtaining that particular set of data, given the chosen probability distribution model.
		It applies to every form of censored or multicensored data, and it is even possible to use the technique across several stress cells and estimate acceleration model parameters at the same time as life distribution parameters.
	www.itl.nist.gov /div898/handbook/apr/section4/apr412.htm (577 words)

	Stat 5102 (Geyer, Spring 2003) MLE
		In order to do maximum likelihood estimation (MLE) using the computer we need to write the likelihood function or log likelihood function (usually the latter) as a function in the computer language we are using.
		function that calculates the minus the log likelihood rather than the log likelihood (stand on your head and maximization becomes minimization).
		The main difference is that the argument to the function must be a vector of parameters.
	www.stat.umn.edu /geyer/5102/examp/rlike.html (1324 words)

	1.3.6.5.2. Maximum Likelihood
		Maximum likelihood estimation begins with the mathematical expression known as a likelihood function of the sample data.
		Loosely speaking, the likelihood of a set of data is the probability of obtaining that particular set of data given the chosen probability model.
		Those values of the parameter that maximize the sample likelihood are known as the maximum likelihood estimates.
	www.itl.nist.gov /div898/handbook/eda/section3/eda3652.htm (583 words)

	Likelihood: theory and application to structure refinement
		The parameters of this likelihood function are the mean and standard deviation we assume for the Gaussian probability distribution.
		In fact, if we have the correct mean the likelihood function will generally balance out the influence of the sparsely-populated tails and the heavily-populated centre to give us the correct standard deviation.
		The likelihood function is the probability of the data given the model (p(data;model), in more formal notation), but we tend to think of it as the probability that the model is a good reflection of reality, given the data (p(model;data)).
	www-structmed.cimr.cam.ac.uk /Course/Likelihood/likelihood.html (5356 words)

	[No title]
		If the detection error is considered Gaussian with covariance matrix R, the maximum likelihood estimate is found by minimising where h is the rendering function (returns the location of the features when projected onto the camera planes given the pose xk).
		The ratio of a likelihood function for an unknown parameter vector to the likelihood function calculated at the estimated parameter vector.
		Two central unifying components of statistics are the likelihood function and the exponential family.
	www.lycos.com /info/likelihood-function--miscellaneous.html (312 words)

	Cone Fit Studies
		The likelihood function is the weighted sum of a gaussian term,
		Each plot shows the function variation with respect to a pair of parameters with the remaining parameters held at their generated values various parameter pairs.
		This result is not surprising since the function depends only on the relative timing of the hits and no assumptions are made about their angular distribution.
	home.fnal.gov /~kasper/boone/scheme4 (1510 words)

	Unimodal Likelihood Function
		This section describes an MMC algorithm suitable for problems where the likelihood function is unimodal and of fixed dimension.
		For the unimodal likelihood function case the minimising MMLD region can be found using Algorithm 1.
		The message lengths and corresponding normalised weights for each link function are given in Table 1.
	www.csse.monash.edu.au /~dld/Publications/2003/node4.html (924 words)

	FAQ: Convergence of ml
		For some likelihoods, these messages are the norm for early iterations, but they typically disappear for the final 4 or 5 iterations.
		Besides collinearity, "nonconcave function encountered" at the last step can happen when the true maximum of the likelihood is at infinity for a parameter.
		Theoretically, this implies that the function is not analytic at b0, but this is usually not the case.
	www.stata.com /support/faqs/stat/ml.html (1817 words)

	Maximum Likelihood Procedures
		In this case, of course, a closed-form solution to the likelihood equations is available, but in general it will be necessary to resort to an iterative non-linear procedure to solve the likelihood equations.
		The log likelihood function is a sum of @(n@) terms, one for each observation.
		The criteria used to determine whether the algorithm has converged depends on a quadratic form in the gradient of the log likelihood function with matrix equal to the negative of the inverse of hessian.
	emlab.berkeley.edu /sst/max.like.html (5326 words)

	Likelihood function - MLpedia
		A likelihood function is a conditional probability P(Y
		X) considered as a function of its second argument X with its first argument Y held fixed.
		where the posterior over a latent variable X can be found by multiplying the prior by the likelihood and renormalising.
	www.mlpedia.org /index.php?title=Likelihood (55 words)

	Roll your own likelihood function with R
		Here are the formulae for the OLS likelihood, and the notation that I use.
		Confucius he said, when you write a likelihood function, do take the trouble of also writing it's gradient (the vector of first derivatives).
		But the OLS likelihood is unique and simple; it is globally quasiconcave and has a clear top.
	www.mayin.org /ajayshah/KB/R/documents/mle/mle.html (757 words)

	[No title]
		Maximum likelihood is simply an estimation procedure—one of many potential ways in which one might generate a guess as to what the true EMBED Equation.3 is. It turns out that maximum likelihood estimates have many desirable properties, especially when the sample size is fairly large.
		The maximum of a well-behaved function is obtained by finding the place where the derivative equals zero and establishing that the second derivative (the derivative of the derivative) is negative.
		The likelihood function for the data is: EMBED Equation.3 Thus, the log-likelihood for the data is: EMBED Equation.3 Based on the first order conditions, it is easy to show that the ML estimator is simply the mean of the EMBED Equation.3 .
	research.yale.edu /vote/pl504/masses05.doc (2617 words)

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