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70-438: The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster , Nan Laird , and Donald Rubin . They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele frequencies by Cedric Smith . Another was proposed by H.O. Hartley in 1958, and Hartley and Hocking in 1977, from which many of

140-406: A `covariance adjustment' to correct the analysis of the M step, capitalising on extra information captured in the imputed complete data". Expectation conditional maximization (ECM) replaces each M step with a sequence of conditional maximization (CM) steps in which each parameter θ i is maximized individually, conditionally on the other parameters remaining fixed. Itself can be extended into

210-426: A data set are missing completely at random (MCAR) if the events that lead to any particular data-item being missing are independent both of observable variables and of unobservable parameters of interest, and occur entirely at random. When data are MCAR, the analysis performed on the data is unbiased; however, data are rarely MCAR. In the case of MCAR, the missingness of data is unrelated to any study variable: thus,

280-432: A first-order auto-regressive process, an updated process noise variance estimate can be calculated by where x ^ k {\displaystyle {\widehat {x}}_{k}} and x ^ k + 1 {\displaystyle {\widehat {x}}_{k+1}} are scalar state estimates calculated by a filter or a smoother. The updated model coefficient estimate

350-436: A hindrance to make effective use of data at scale, including through both classical statistical and current machine learning methods. For example, there might be bias inherent in the reasons why some data might be missing in patterns, which might have implications in predictive fairness for machine learning models. Furthermore, established methods for dealing with missing data, such as imputation , do not usually take into account

420-435: A larger number of imputations. However, a too-small number of imputations can lead to a substantial loss of statistical power , and some scholars now recommend 20 to 100 or more. Any multiply-imputed data analysis must be repeated for each of the imputed data sets and, in some cases, the relevant statistics must be combined in a relatively complicated way. Multiple imputation is not conducted in specific disciplines, as there

490-399: A local maximum, such as random-restart hill climbing (starting with several different random initial estimates θ ( t ) {\displaystyle {\boldsymbol {\theta }}^{(t)}} ), or applying simulated annealing methods. EM is especially useful when the likelihood is an exponential family , see Sundberg (2019, Ch. 8) for a comprehensive treatment:

560-458: A local minimum of the cost function. Although an EM iteration does increase the observed data (i.e., marginal) likelihood function, no guarantee exists that the sequence converges to a maximum likelihood estimator . For multimodal distributions , this means that an EM algorithm may converge to a local maximum of the observed data likelihood function, depending on starting values. A variety of heuristic or metaheuristic approaches exist to escape

630-867: A posteriori (MAP) estimates for Bayesian inference in the original paper by Dempster, Laird, and Rubin. Other methods exist to find maximum likelihood estimates, such as gradient descent , conjugate gradient , or variants of the Gauss–Newton algorithm . Unlike EM, such methods typically require the evaluation of first and/or second derivatives of the likelihood function. Expectation-Maximization works to improve Q ( θ ∣ θ ( t ) ) {\displaystyle Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})} rather than directly improving log ⁡ p ( X ∣ θ ) {\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})} . Here it

700-399: A sample of n {\displaystyle n} independent observations from a mixture of two multivariate normal distributions of dimension d {\displaystyle d} , and let z = ( z 1 , z 2 , … , z n ) {\displaystyle \mathbf {z} =(z_{1},z_{2},\ldots ,z_{n})} be

770-547: A set of unobserved latent data or missing values Z {\displaystyle \mathbf {Z} } , and a vector of unknown parameters θ {\displaystyle {\boldsymbol {\theta }}} , along with a likelihood function L ( θ ; X , Z ) = p ( X , Z ∣ θ ) {\displaystyle L({\boldsymbol {\theta }};\mathbf {X} ,\mathbf {Z} )=p(\mathbf {X} ,\mathbf {Z} \mid {\boldsymbol {\theta }})} ,

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840-470: A subset of variables from the union of measurement modalities. In these situations, missing values may relate to the various sampling methodologies used to collect the data or reflect characteristics of the wider population of interest, and so may impart useful information. For instance, in a health context, structured missingness has been observed as a consequence of linking clinical, genomic and imaging data. The presence of structured missingness may be

910-417: A test for refuting MAR/MCAR reads as follows: For any three variables X,Y , and Z where Z is fully observed and X and Y partially observed, the data should satisfy: X ⊥ ⊥ R y | ( R x , Z ) {\displaystyle X\perp \!\!\!\perp R_{y}|(R_{x},Z)} . In words, the observed portion of X should be independent on

980-462: Is a lack of training or misconceptions about them. Methods such as listwise deletion have been used to impute data but it has been found to introduce additional bias. There is a beginner guide that provides a step-by-step instruction how to impute data.   The expectation-maximization algorithm is an approach in which values of the statistics which would be computed if a complete dataset were available are estimated (imputed), taking into account

1050-467: Is a type of missingness that can occur in longitudinal studies—for instance studying development where a measurement is repeated after a certain period of time. Missingness occurs when participants drop out before the test ends and one or more measurements are missing. Data often are missing in research in economics , sociology , and political science because governments or private entities choose not to, or fail to, report critical statistics, or because

1120-494: Is also possible to consider the EM algorithm as a subclass of the MM (Majorize/Minimize or Minorize/Maximize, depending on context) algorithm, and therefore use any machinery developed in the more general case. The Q-function used in the EM algorithm is based on the log likelihood. Therefore, it is regarded as the log-EM algorithm. The use of the log likelihood can be generalized to that of

1190-545: Is an indicator function and f {\displaystyle f} is the probability density function of a multivariate normal. Arthur P. Dempster Arthur Pentland Dempster (born 1929) is a Professor Emeritus in the Harvard University Department of Statistics. He was one of four faculty when the department was founded in 1957. Dempster received his B.A. in mathematics and physics (1952) and M.A. in mathematics (1953), both from

1260-478: Is an exact generalization of the log-EM algorithm. No computation of gradient or Hessian matrix is needed. The α-EM shows faster convergence than the log-EM algorithm by choosing an appropriate α. The α-EM algorithm leads to a faster version of the Hidden Markov model estimation algorithm α-HMM. EM is a partially non-Bayesian, maximum likelihood method. Its final result gives a probability distribution over

1330-492: Is applied use Z {\displaystyle \mathbf {Z} } as a latent variable indicating membership in one of a set of groups: However, it is possible to apply EM to other sorts of models. The motivation is as follows. If the value of the parameters θ {\displaystyle {\boldsymbol {\theta }}} is known, usually the value of the latent variables Z {\displaystyle \mathbf {Z} } can be found by maximizing

1400-403: Is complete information. Since MAR is an assumption that is impossible to verify statistically, we must rely on its substantive reasonableness. An example is that males are less likely to fill in a depression survey but this has nothing to do with their level of depression, after accounting for maleness. Depending on the analysis method, these data can still induce parameter bias in analyses due to

1470-428: Is observed regardless of the status of X . Moreover, in order to obtain a consistent estimate it is crucial that the first term be P ( X | Y ) {\displaystyle P(X|Y)} as opposed to P ( Y | X ) {\displaystyle P(Y|X)} . In many cases model based techniques permit the model structure to undergo refutation tests. Any model which implies

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1540-452: Is obtained via The convergence of parameter estimates such as those above are well studied. A number of methods have been proposed to accelerate the sometimes slow convergence of the EM algorithm, such as those using conjugate gradient and modified Newton's methods (Newton–Raphson). Also, EM can be used with constrained estimation methods. Parameter-expanded expectation maximization (PX-EM) algorithm often provides speed up by "us[ing]

1610-399: Is shown that improvements to the former imply improvements to the latter. For any Z {\displaystyle \mathbf {Z} } with non-zero probability p ( Z ∣ X , θ ) {\displaystyle p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }})} , we can write We take the expectation over possible values of

1680-438: Is stored for the variable in an observation . Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data. Missing data can occur because of nonresponse: no information is provided for one or more items or for a whole unit ("subject"). Some items are more likely to generate a nonresponse than others: for example items about private subjects such as income. Attrition

1750-616: Is the expectation of a constant, so we get: where H ( θ ∣ θ ( t ) ) {\displaystyle H({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})} is defined by the negated sum it is replacing. This last equation holds for every value of θ {\displaystyle {\boldsymbol {\theta }}} including θ = θ ( t ) {\displaystyle {\boldsymbol {\theta }}={\boldsymbol {\theta }}^{(t)}} , and subtracting this last equation from

1820-596: Is the partially overlapping samples t-test. This is valid under normality and assuming MCAR Methods which involve reducing the data available to a dataset having no missing values include: Methods which take full account of all information available, without the distortion resulting from using imputed values as if they were actually observed: Partial identification methods may also be used. Model based techniques, often using graphs, offer additional tools for testing missing data types (MCAR, MAR, MNAR) and for estimating parameters under missing data conditions. For example,

1890-406: Is your salary?’, analyses that do not take into account this missing at random (MAR pattern (see below)) may falsely fail to find a positive association between IQ and salary. Because of these problems, methodologists routinely advise researchers to design studies to minimize the occurrence of missing values. Graphical models can be used to describe the missing data mechanism in detail. Values in

1960-521: The Expectation conditional maximization either (ECME) algorithm. This idea is further extended in generalized expectation maximization (GEM) algorithm, in which is sought only an increase in the objective function F for both the E step and M step as described in the As a maximization–maximization procedure section. GEM is further developed in a distributed environment and shows promising results. It

2030-531: The University of Toronto . He obtained his Ph.D. in mathematical statistics from Princeton University in 1956. His thesis, titled The two-sample multivariate problem in the degenerate case , was written under the supervision of John Tukey . Among his contributions to statistics are the Dempster–Shafer theory and the expectation-maximization (EM) algorithm . Dempster was a Putnam Fellow in 1951. He

2100-442: The exponential family , as claimed by Dempster–Laird–Rubin. The EM algorithm is used to find (local) maximum likelihood parameters of a statistical model in cases where the equations cannot be solved directly. Typically these models involve latent variables in addition to unknown parameters and known data observations. That is, either missing values exist among the data, or the model can be formulated more simply by assuming

2170-404: The maximum likelihood calculation where x ^ k {\displaystyle {\widehat {x}}_{k}} are scalar output estimates calculated by a filter or a smoother from N scalar measurements z k {\displaystyle z_{k}} . The above update can also be applied to updating a Poisson measurement noise intensity. Similarly, for

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2240-494: The maximum likelihood estimate (MLE) of the unknown parameters is determined by maximizing the marginal likelihood of the observed data However, this quantity is often intractable since Z {\displaystyle \mathbf {Z} } is unobserved and the distribution of Z {\displaystyle \mathbf {Z} } is unknown before attaining θ {\displaystyle {\boldsymbol {\theta }}} . The EM algorithm seeks to find

2310-506: The E step and the M step are interpreted as projections under dual affine connections , called the e-connection and the m-connection; the Kullback–Leibler divergence can also be understood in these terms. Let x = ( x 1 , x 2 , … , x n ) {\displaystyle \mathbf {x} =(\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n})} be

2380-468: The E step becomes the sum of expectations of sufficient statistics , and the M step involves maximizing a linear function. In such a case, it is usually possible to derive closed-form expression updates for each step, using the Sundberg formula (proved and published by Rolf Sundberg, based on unpublished results of Per Martin-Löf and Anders Martin-Löf ). The EM method was modified to compute maximum

2450-461: The components to have zero variance and the mean parameter for the same component to be equal to one of the data points. The convergence of expectation-maximization (EM)-based algorithms typically requires continuity of the likelihood function with respect to all the unknown parameters (referred to as optimization variables). Given the statistical model which generates a set X {\displaystyle \mathbf {X} } of observed data,

2520-403: The conclusions drawn about the population. Some data analysis techniques are not robust to missingness, and require to "fill in", or impute the missing data. Rubin (1987) argued that repeating imputation even a few times (5 or less) enormously improves the quality of estimation. For many practical purposes, 2 or 3 imputations capture most of the relative efficiency that could be captured with

2590-429: The contingent emptiness of cells (male, very high depression may have zero entries). However, if the parameter is estimated with Full Information Maximum Likelihood, MAR will provide asymptotically unbiased estimates. Missing not at random (MNAR) (also known as nonignorable nonresponse) is data that is neither MAR nor MCAR (i.e. the value of the variable that's missing is related to the reason it's missing). To extend

2660-437: The derivative of the likelihood is (arbitrarily close to) zero at that point, which in turn means that the point is either a local maximum or a saddle point . In general, multiple maxima may occur, with no guarantee that the global maximum will be found. Some likelihoods also have singularities in them, i.e., nonsensical maxima. For example, one of the solutions that may be found by EM in a mixture model involves setting one of

2730-413: The existence of further unobserved data points. For example, a mixture model can be described more simply by assuming that each observed data point has a corresponding unobserved data point, or latent variable, specifying the mixture component to which each data point belongs. Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all

2800-426: The experimenters can control the level of missingness, and prevent missing values before gathering the data. For example, in computer questionnaires, it is often not possible to skip a question. A question has to be answered, otherwise one cannot continue to the next. So missing values due to the participant are eliminated by this type of questionnaire, though this method may not be permitted by an ethics board overseeing

2870-457: The factorized Q approximation as described above ( variational Bayes ), solving can iterate over each latent variable (now including θ ) and optimize them one at a time. Now, k steps per iteration are needed, where k is the number of latent variables. For graphical models this is easy to do as each variable's new Q depends only on its Markov blanket , so local message passing can be used for efficient inference. In information geometry ,

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2940-418: The function: where q is an arbitrary probability distribution over the unobserved data z and H(q) is the entropy of the distribution q . This function can be written as where p Z ∣ X ( ⋅ ∣ x ; θ ) {\displaystyle p_{Z\mid X}(\cdot \mid x;\theta )} is the conditional distribution of the unobserved data given

3010-520: The ideas in the Dempster–Laird–Rubin paper originated. Another one by S.K Ng, Thriyambakam Krishnan and G.J McLachlan in 1977. Hartley’s ideas can be broadened to any grouped discrete distribution. A very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers, following his collaboration with Per Martin-Löf and Anders Martin-Löf . The Dempster–Laird–Rubin paper in 1977 generalized

3080-488: The independence between a partially observed variable X and the missingness indicator of another variable Y (i.e. R y {\displaystyle R_{y}} ), conditional on R x {\displaystyle R_{x}} can be submitted to the following refutation test: X ⊥ ⊥ R y | R x = 0 {\displaystyle X\perp \!\!\!\perp R_{y}|R_{x}=0} . Finally,

3150-448: The information is not available. Sometimes missing values are caused by the researcher—for example, when data collection is done improperly or mistakes are made in data entry. These forms of missingness take different types, with different impacts on the validity of conclusions from research: Missing completely at random, missing at random, and missing not at random. Missing data can be handled similarly as censored data . Understanding

3220-429: The kinds of people who will still refuse or remain unreachable after additional effort. In situations where missing values are likely to occur, the researcher is often advised on planning to use methods of data analysis methods that are robust to missingness. An analysis is robust when we are confident that mild to moderate violations of the technique's key assumptions will produce little or no bias , or distortion in

3290-485: The latent variables (in the Bayesian style) together with a point estimate for θ (either a maximum likelihood estimate or a posterior mode). A fully Bayesian version of this may be wanted, giving a probability distribution over θ and the latent variables. The Bayesian approach to inference is simply to treat θ as another latent variable. In this paradigm, the distinction between the E and M steps disappears. If using

3360-458: The latent variables that determine the component from which the observation originates. where The aim is to estimate the unknown parameters representing the mixing value between the Gaussians and the means and covariances of each: where the incomplete-data likelihood function is and the complete-data likelihood function is or where I {\displaystyle \mathbb {I} }

3430-485: The log-likelihood over all possible values of Z {\displaystyle \mathbf {Z} } , either simply by iterating over Z {\displaystyle \mathbf {Z} } or through an algorithm such as the Viterbi algorithm for hidden Markov models . Conversely, if we know the value of the latent variables Z {\displaystyle \mathbf {Z} } , we can find an estimate of

3500-834: The maximum likelihood estimate of the marginal likelihood by iteratively applying these two steps: More succinctly, we can write it as one equation: θ ( t + 1 ) = a r g m a x θ E Z ∼ p ( ⋅ | X , θ ( t ) ) ⁡ [ log ⁡ p ( X , Z | θ ) ] {\displaystyle {\boldsymbol {\theta }}^{(t+1)}={\underset {\boldsymbol {\theta }}{\operatorname {arg\,max} }}\operatorname {E} _{\mathbf {Z} \sim p(\cdot |\mathbf {X} ,{\boldsymbol {\theta }}^{(t)})}\left[\log p(\mathbf {X} ,\mathbf {Z} |{\boldsymbol {\theta }})\right]\,} The typical models to which EM

3570-434: The method and sketched a convergence analysis for a wider class of problems. The Dempster–Laird–Rubin paper established the EM method as an important tool of statistical analysis. See also Meng and van Dyk (1997). The convergence analysis of the Dempster–Laird–Rubin algorithm was flawed and a correct convergence analysis was published by C. F. Jeff Wu in 1983. Wu's proof established the EM method's convergence also outside of

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3640-453: The missingness status of Y, conditional on every value of Z . Failure to satisfy this condition indicates that the problem belongs to the MNAR category. (Remark: These tests are necessary for variable-based MAR which is a slight variation of event-based MAR. ) When data falls into MNAR category techniques are available for consistently estimating parameters when certain conditions hold in

3710-425: The model. For example, if Y explains the reason for missingness in X and Y itself has missing values, the joint probability distribution of X and Y can still be estimated if the missingness of Y is random. The estimand in this case will be: where R x = 0 {\displaystyle R_{x}=0} and R y = 0 {\displaystyle R_{y}=0} denote

3780-478: The observation that there is a way to solve these two sets of equations numerically. One can simply pick arbitrary values for one of the two sets of unknowns, use them to estimate the second set, then use these new values to find a better estimate of the first set, and then keep alternating between the two until the resulting values both converge to fixed points. It's not obvious that this will work, but it can be proven in this context. Additionally, it can be proven that

3850-447: The observed data x {\displaystyle x} and D K L {\displaystyle D_{KL}} is the Kullback–Leibler divergence . Then the steps in the EM algorithm may be viewed as: A Kalman filter is typically used for on-line state estimation and a minimum-variance smoother may be employed for off-line or batch state estimation. However, these minimum-variance solutions require estimates of

3920-463: The observed portions of their respective variables. Different model structures may yield different estimands and different procedures of estimation whenever consistent estimation is possible. The preceding estimand calls for first estimating P ( X | Y ) {\displaystyle P(X|Y)} from complete data and multiplying it by P ( Y ) {\displaystyle P(Y)} estimated from cases in which Y

3990-585: The parameters θ {\displaystyle {\boldsymbol {\theta }}} fairly easily, typically by simply grouping the observed data points according to the value of the associated latent variable and averaging the values, or some function of the values, of the points in each group. This suggests an iterative algorithm, in the case where both θ {\displaystyle {\boldsymbol {\theta }}} and Z {\displaystyle \mathbf {Z} } are unknown: The algorithm as just described monotonically approaches

4060-427: The participants with completely observed data are in effect a random sample of all the participants assigned a particular intervention. With MCAR, the random assignment of treatments is assumed to be preserved, but that is usually an unrealistically strong assumption in practice. Missing at random (MAR) occurs when the missingness is not random, but where missingness can be fully accounted for by variables where there

4130-419: The pattern of missing data. In this approach, values for individual missing data-items are not usually imputed. In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In the comparison of two paired samples with missing data, a test statistic that uses all available data without the need for imputation

4200-509: The place of missing data, (2) omission —where samples with invalid data are discarded from further analysis and (3) analysis —by directly applying methods unaffected by the missing values. One systematic review addressing the prevention and handling of missing data for patient-centered outcomes research identified 10 standards as necessary for the prevention and handling of missing data. These include standards for study design, study conduct, analysis, and reporting. In some practical application,

4270-1098: The previous equation gives However, Gibbs' inequality tells us that H ( θ ∣ θ ( t ) ) ≥ H ( θ ( t ) ∣ θ ( t ) ) {\displaystyle H({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})\geq H({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})} , so we can conclude that In words, choosing θ {\displaystyle {\boldsymbol {\theta }}} to improve Q ( θ ∣ θ ( t ) ) {\displaystyle Q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})} causes log ⁡ p ( X ∣ θ ) {\displaystyle \log p(\mathbf {X} \mid {\boldsymbol {\theta }})} to improve at least as much. The EM algorithm can be viewed as two alternating maximization steps, that is, as an example of coordinate descent . Consider

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4340-859: The previous example, this would occur if men failed to fill in a depression survey because of their level of depression. Samuelson and Spirer (1992) discussed how missing and/or distorted data about demographics, law enforcement, and health could be indicators of patterns of human rights violations. They gave several fairly well documented examples. Missing data can also arise in subtle ways that are not well accounted for in classical theory. An increasingly encountered problem arises in which data may not be MAR but missing values exhibit an association or structure, either explicitly or implicitly. Such missingness has been described as ‘structured missingness’. Structured missingness commonly arises when combining information from multiple studies, each of which may vary in its design and measurement set and therefore only contain

4410-405: The reasons why data are missing is important for handling the remaining data correctly. If values are missing completely at random, the data sample is likely still representative of the population. But if the values are missing systematically, analysis may be biased. For example, in a study of the relation between IQ and income, if participants with an above-average IQ tend to skip the question ‘What

4480-508: The research. In survey research, it is common to make multiple efforts to contact each individual in the sample, often sending letters to attempt to persuade those who have decided not to participate to change their minds. However, such techniques can either help or hurt in terms of reducing the negative inferential effects of missing data, because the kind of people who are willing to be persuaded to participate after initially refusing or not being home are likely to be significantly different from

4550-431: The state-space model parameters. EM algorithms can be used for solving joint state and parameter estimation problems. Filtering and smoothing EM algorithms arise by repeating this two-step procedure: Suppose that a Kalman filter or minimum-variance smoother operates on measurements of a single-input-single-output system that possess additive white noise. An updated measurement noise variance estimate can be obtained from

4620-512: The structure of the missing data and so development of new formulations is needed to deal with structured missingness appropriately or effectively. Finally, characterising structured missingness within the classical framework of MCAR, MAR, and MNAR is a work in progress. Missing data reduces the representativeness of the sample and can therefore distort inferences about the population. Generally speaking, there are three main approaches to handle missing data: (1) Imputation —where values are filled in

4690-540: The unknown data Z {\displaystyle \mathbf {Z} } under the current parameter estimate θ ( t ) {\displaystyle \theta ^{(t)}} by multiplying both sides by p ( Z ∣ X , θ ( t ) ) {\displaystyle p(\mathbf {Z} \mid \mathbf {X} ,{\boldsymbol {\theta }}^{(t)})} and summing (or integrating) over Z {\displaystyle \mathbf {Z} } . The left-hand side

4760-469: The unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations. In statistical models with latent variables, this is usually impossible. Instead, the result is typically a set of interlocking equations in which the solution to the parameters requires the values of the latent variables and vice versa, but substituting one set of equations into the other produces an unsolvable equation. The EM algorithm proceeds from

4830-483: The α-log likelihood ratio. Then, the α-log likelihood ratio of the observed data can be exactly expressed as equality by using the Q-function of the α-log likelihood ratio and the α-divergence. Obtaining this Q-function is a generalized E step. Its maximization is a generalized M step. This pair is called the α-EM algorithm which contains the log-EM algorithm as its subclass. Thus, the α-EM algorithm by Yasuo Matsuyama

4900-483: Was elected as an American Statistical Association Fellow in 1964, an Institute of Mathematical Statistics Fellow in 1963, and an American Academy of Arts and Sciences Fellow in 1997. This article about a statistician from the United States is a stub . You can help Misplaced Pages by expanding it . Missing values In statistics , missing data , or missing values , occur when no data value

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