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Topic: Maximum likelihood

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Likelihood

Principle of maximum entropy

Likelihood principle

Posterior probability

Likelihood ratio test

Probability density function

Normal distribution

Estimation of covariance matrices

Conditional probability

Multivariate normal distribution

Bayes theorem

Logit

Random variable

Sunrise problem

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	Stats: Maximum likelihood estimation (May 6, 2003)
		Maximum likelihood is an approach that looks at a large class of distributions and then chooses the "best" distribution.
		The log of the likelihood function often simplifies many of the calculations, and if you find the maximum of the log likelihood that also has to be the maximum of the likelihood itself.
		I won't show all the equations, but the maximum likelihood estimate of mu ends up equaling the sample mean and the maximum likelihood estimate of sigma ends up equaling, not the sample standard deviation exactly, but something very close where you replace n-1 with n in the formula.
	www.childrensmercy.org /stats/ask/mle.asp (642 words)

	NationMaster - Encyclopedia: Maximum likelihood
		Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set.
		Likelihood of different proportion parameter values for a binomial process with t = 3 and n = 10; the ML estimator occurs at the mode with the peak (maximum) of the curve.
		When maximising the likelihood, we may equivalently maximise the log of the likelihood, since log is a continuous strictly increasing function over the range of the likelihood.
	www.nationmaster.com /encyclopedia/Maximum_likelihood (2910 words)

	NationMaster - Encyclopedia: Method of maximum likelihood
		In statistics, the method of maximum likelihood, pioneered by geneticist and statistician Sir Ronald A. Fisher, is a method of point estimation, that uses as an estimate of an unobservable population parameter the member of the parameter space that maximizes the likelihood function.
		Compared with the attainable maximum size of about 30 mm, the growth of juveniles is extremely slow because their growth is less susceptible to environmental factors until the second winter.
		Likelihood function The location and scale parameters at the first sampling ([a.sub.0] and [b.sub.0]), the coefficients of Equation 6 ([s.sub.max], [[alpha].sub.a] and [[beta].sub.k]), and the coefficients of Equations 7 and 8 ([[gamma].sub.1] and [[gamma].sub.2]) are estimated as values that maximize total log-likelihood.
	www.nationmaster.com /encyclopedia/Method-of-maximum-likelihood (640 words)

	Maximum likelihood - Christoph's Personal Wiki
		Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set.
		The maximum likelihood estimator may not be unique, or indeed may not even exist.
		Maximum likelihood estimators achieve minimum variance (as given by the Cramer-Rao lower bound) in the limit as the sample size tends to infinity.
	wiki.christophchamp.com /index.php/Maximum_likelihood (648 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.
		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.
		are those that yield the highest value of (i.e., the maximum) of the likelihood (and log-likelihood) function.
	am.air.org /help/NAEPTextbook/htm/dmarginalmaximumlikelihood.htm (493 words)

	BGIM : Maximum Likelihood Estimation Primer
		The aim of maximum likelihood estimation is to find the parameter value(s) that makes the observed data most likely.
		This is because the likelihood of the parameters given the data is defined to be equal to the probability of the data given the parameters
		The likelihood framework conceptually takes all of this in its stride, however, and this is what makes it the work-horse of many modern statistical methods.
	statgen.iop.kcl.ac.uk /bgim/mle/sslike_3.html (658 words)

	Maximum Likelihood Estimation of K Distribution Parameters for SAR Data -- from Mathematica Information Center
		In this paper, we apply a maximum likelihood estimation method directly to the K distribution.
		We investigate the accuracy and uncertainties in maximum likelihood parameter estimates as functions of sample size and the parameters themselves.
		We also compare our results with those from a new method given by Raghavan and from a nonstandard method of moments technique; maximum likelihood parameter estimates prove to be at least as accurate as those from the other estimators in all cases tested, and are more accurate in most cases.
	library.wolfram.com /infocenter/Articles/1142 (297 words)

	[No title]
		An common alternative to the least squares loss function is to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function; the term maximum likelihood was first used by Fisher, 1922a).
		The method of maximum likelihood (the term first used by Fisher, 1922a) is a general method of estimating parameters of a population by values that maximize the likelihood (L) of a sample.
		The maximum unconfounding criterion specifies that design generators should be chosen such that the maximum number of interactions of less than or equal to the crucial order, given the resolution, are unconfounded with all other interactions of the crucial order.
	www.statsoft.com /textbook/glosm.html (5420 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.
		Except for a few cases where the maximum likelihood formulas are in fact simple, it is generally best to rely on high quality statistical software to obtain maximum likelihood estimates.
	www.itl.nist.gov /div898/handbook/eda/section3/eda3652.htm (583 words)

	11.7 Maximum Likelihood Classifier (Site not responding. Last check: )
		The maximum likelihood classifier is one of the most popular methods of classification in remote sensing, in which a pixel with the maximum likelihood is classified into the corresponding class.
		The likelihood Lk is defined as the posterior probability of a pixel belonging to class k.
		In the case where the variance-covariance matrix is symmetric, the likelihood is the same as the Euclidian distance, while in case where the determinants are equal each other, the likelihood becomes the same as the Mahalanobis distances.
	www.profc.udec.cl /~gabriel/tutoriales/rsnote/cp11/cp11-7.htm (304 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/Likelihood.html (203 words)

	Closed-Form Maximum Likelihood Estimates of Nearest Neighbor Spatial Dependence
		We report likelihood ratio statistics (i.e., twice the difference between the restricted and unrestricted log-likelihoods) for the null hypothesis that a variable has no effect or in many cases that a variable and its spatial lag have no effect.
		As one might expect, the nearest neighbor maximum likelihood estimates usually fell between the OLS and maximum likelihood using the four nearest neighbors.
		The log-likelihood was –6307.1 for OLS without lagged independent variables, -5964.6 for maximum likelihood with the nearest neighbor, -5954.9 for maximum likelihood with the nearest neighbor and the third-order neighbor, and –5678.8 for maximum likelihood with four nearest neighbors.
	www.spatial-statistics.com /pace_manuscripts/closest_neighbors/html/nearest_neighbor.html (5728 words)

	Maximum Likelihood
		Maximum Likelihood is a method for the inference of phylogeny.
		To calculate the likelihood for site j, we have to consider all the possible scenarios by which the nucleotides present at the tips of the tree could have evolved.
		Since the individual likelihoods are extremely small numbers it is convenient to sum the log likelihoods at each site and report the likelihood of the entire tree as the log likelihood.
	www.icp.ucl.ac.be /~opperd/private/max_likeli.html (774 words)

	Phylogenetic Analysis Using Maximum Likelihood
		Under maximum likelihood, the best estimate of the parameter values is taken to be those that give the highest probability of producing the observed outcome.
		The maximum likelihood estimate of p is then the value which gives the highest probability of producing h=59 (in the present case that would be the rather obvious estimate p=0.59).
		In phylogenetic analysis using maximum likelihood, the observed data is most often taken to be the set of aligned sequences.
	www.cbs.dtu.dk /courses/PR/likelihood.php (2729 words)

	Maximum Likelihood
		The three main components of the statistical approach are (i) the data, (ii) a model describing the probability of observing the data, and (iii) a criterion that allows us to move from the data and model to an estimate of the parameters of the model.
		The maximum likelihood method finds the estimate of a parameter that maximizes the probability of observing the data given a specific model for the data.
		Here, the likelihood calculated under the null hypothesis is in the numerator and the likelihood calculated under the alternative hypothesis is in the denominator.
	www.sciencemag.org /feature/data/phylo/coin/coin.html (1384 words)

	The Likelihood Function
		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.
		For nice (defined as the opposite of naughty) likelihoods (like the binomial), it turns out that there is a single value of p, given the data, that is most likely; that is, makes the likelihood as big as possible.
	fisher.forestry.uga.edu /popdyn/Likelihood.html (669 words)

	Maximum Likelihood Estimator of the 85th
		The log likelihood ln[f(s)]+ ln[1-F(s)] is the likelihood of a pair driving at the same speed s, presumably because the faster driver is obstructed by the slower.
		Solving for the maximum likelihood estimators of the mean and variance explicitly appears to be difficult.
		Table 2 shows the maximum likelihood estimates of the parameters for one simulation in which all 200 vehicle speeds were observed, blocked or unobstructed.
	home.comcast.net /~pstlarry/Speed.htm (686 words)

	Maximum Likelihood
		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
		The maximum likelihood under the null hypothesis is logL0 = -7628.025.
		The maximum likelihood under the null hypothesis is logL0 = -7585.343.
	ucjeps.berkeley.edu /bryolab/ib200/maximumlikelihood.html (2941 words)

	MLE (Maximum Likelihood) Parameter Estimation
		The idea behind maximum likelihood parameter estimation is to determine the parameters that maximize the probability (likelihood) of the sample data.
		From a statistical point of view, the method of maximum likelihood is considered to be more robust (with some exceptions) and yields estimators with good statistical properties.
		Although the methodology for maximum likelihood estimation is simple, the implementation is mathematically intense.
	www.weibull.com /AccelTestWeb/mle_maximum_likelihood_parameter_estimation.htm (665 words)

	dnamlk
		This program implements the maximum likelihood method for DNA sequences under the constraint that the trees estimated must be consistent with a molecular clock.
		These empirical frequencies are not really the maximum likelihood estimates of the base frequencies, but they will often be close to those values (what they are is maximum likelihood estimates under a "star" or "explosion" phylogeny).
		Thus the assessment of 95% of the likelihood, in tabulating the ancestral states, refers to 95% of the likelihood that is accounted for by that particular combination of rates.
	evolution.genetics.washington.edu /phylip/doc/dnamlk.html (4757 words)

	Likelihood: theory and application to structure refinement
		Since maximum likelihood refinement is based directly on these probabilities, the course of refinement depends very sensitively on having good estimates.
		The principle of maximum likelihood is quite intuitive, and it leads to useful results, but you might argue that there is a logical problem.
		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)

	Statistics 5102 (Geyer, Spring 2007) Examples: Maximum Likelihood Estimation
		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.
		In our particular problem, maximum likelihood for the shape parameter of the gamma distribution, a good estimate of the shape parameter α is the sample mean, since the theoretical mean of the gamma distribution is α / β where β is the
		Maximum likelihood is the only well-known method that is not computer intensive.
	www.stat.umn.edu /geyer/5102/examp/rlike.html (1972 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.
		Maximum likelihood estimation is a totally analytic maximization procedure.
	www.itl.nist.gov /div898/handbook/apr/section4/apr412.htm (577 words)

	Comparison of maximum likelihood and least-squares methods
		To compare maximum likelihood and least-squares residuals we can analyse the behaviour of their gradients.
		But at the end stages of refinement when the model is complete and has a small error then the maximum likelihood could be approximated by the least-squares (see Murshudov et al 1997).
		In the maximum likelihood case for the gradient calculations a map with coefficients is calculated:
	www.ysbl.york.ac.uk /~garib/likelihood/node3.html (247 words)

	Maximum Likelihood
		Likelihood of this tree = (0.000360 * 0.017301 * 0.013841 * 0.013841 * 0.013841 * 0.017301 * 0.013841 * 0.013841 * 0.013841 * 0.017301 * 0.013841 * 0.013841 0.013841 * 0.017301 * 0.013841 * 0.013841 * 0.013841)
		Since, in choosing the likelihood approach, you have adopted a statistical means for making inferences, this is not a trivial question.
		The likelihood ratio test for comparing trees of different shape and with different branch lengths is not as straightforward as the one above for diferent models.
	research.amnh.org /~siddall/methods/ml.html (2405 words)

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