Misplaced Pages

#174825

54-450: A cure is a completely effective treatment for a disease. Cure , or similar, may also refer to: Cure An incurable disease may or may not be a terminal illness ; conversely, a curable illness can still result in the patient's death. The proportion of people with a disease that are cured by a given treatment, called the cure fraction or cure rate , is determined by comparing disease-free survival of treated people against

108-408: 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

162-400: 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 the ideas in

216-462: A cure. Other diseases may prove to have multiple plateaus, so that what was once hailed as a "cure" results unexpectedly in very late relapses. Consequently, patients, parents and psychologists developed the notion of psychological cure , or the moment at which the patient decides that the treatment was sufficiently likely to be a cure as to be called a cure. For example, a patient may declare himself to be "cured", and to determine to live his life as if

270-432: 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

324-399: 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:

378-458: 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

432-416: A matched control group that never had the disease. Another way of determining the cure fraction and/or "cure time" is by measuring when the hazard rate in a diseased group of individuals returns to the hazard rate measured in the general population. Inherent in the idea of a cure is the permanent end to the specific instance of the disease. When a person has the common cold , and then recovers from it,

486-401: A maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. It can be used, for example, to estimate a mixture of gaussians , or to solve the multiple linear regression problem. The EM algorithm was explained and given its name in

540-867: 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

594-399: 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

SECTION 10

#1732851384175

648-547: 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 }})} ,

702-489: 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

756-405: Is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models , where the model depends on unobserved latent variables . The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and

810-471: 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

864-492: 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

918-503: Is necessary to wait before declaring an asymptomatic individual to be cured. Several cure rate models exist, such as the expectation-maximization algorithm and Markov chain Monte Carlo model. It is possible to use cure rate models to compare the efficacy of different treatments. Generally, the survival curves are adjusted for the effects of normal aging on mortality, especially when diseases of older people are being studied. From

972-457: 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]

1026-401: 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

1080-616: 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

1134-448: Is the proportion that are permanently cured, and S ∗ ( t ) {\displaystyle S^{*}(t)} is an exponential curve that represents the survival of the non-cured people. Cure rate curves can be determined through an analysis of the data. The analysis allows the statistician to determine the proportion of people that are permanently cured by a given treatment, and also how long after treatment it

SECTION 20

#1732851384175

1188-465: Is usually made through the Kaplan-Meier estimator approach. The simplest cure rate model was published by Joseph Berkson and Robert P. Gage in 1952. In this model, the survival at any given time is equal to those that are cured plus those that are not cured, but who have not yet died or, in the case of diseases that feature asymptomatic remissions, have not yet re-developed signs and symptoms of

1242-515: 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

1296-443: 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

1350-404: 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

1404-494: 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

1458-517: 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

1512-506: 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

1566-470: 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

1620-462: 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,

1674-421: The cure were definitely confirmed, immediately after treatment. Cures can take the form of natural antibiotics (for bacterial infections ), synthetic antibiotics such as the sulphonamides , or fluoroquinolones , antivirals (for a very few viral infections ), antifungals , antitoxins , vitamins , gene therapy , surgery, chemotherapy, radiotherapy, and so on. Despite a number of cures being developed,

Cure (disambiguation) - Misplaced Pages Continue

1728-438: 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

1782-475: The disease. When all of the non-cured people have died or re-developed the disease, only the permanently cured members of the population will remain, and the DFS curve will be perfectly flat. The earliest point in time that the curve goes flat is the point at which all remaining disease-free survivors are declared to be permanently cured. If the curve never goes flat, then the disease is formally considered incurable (with

1836-413: 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

1890-438: The existing treatments). The Berkson and Gage equation is S ( t ) = p + [ ( 1 − p ) × S ∗ ( t ) ] {\displaystyle S(t)=p+[(1-p)\times S^{*}(t)]} where S ( t ) {\displaystyle S(t)} is the proportion of people surviving at any given point in time, p {\displaystyle p}

1944-457: 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 ,

1998-418: 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

2052-480: 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

2106-454: 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} }

2160-454: The list of incurable diseases remains long. Scurvy became curable (as well as preventable) with doses of vitamin C (for example, in limes) when James Lind published A Treatise on the Scurvy (1753). Antitoxins to diphtheria and tetanus toxins were produced by Emil Adolf von Behring and his colleagues from 1890 onwards. The use of diphtheria antitoxin for the treatment of diphtheria

2214-480: 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

Cure (disambiguation) - Misplaced Pages Continue

2268-834: 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

2322-435: 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

2376-478: 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

2430-447: 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

2484-585: 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

2538-665: The person is said to be cured , even though the person might someday catch another cold. Conversely, a person that has successfully managed a disease, such as diabetes mellitus , so that it produces no undesirable symptoms for the moment, but without actually permanently ending it, is not cured. Related concepts, whose meaning can differ, include response , remission and recovery . In complex diseases, such as cancer, researchers rely on statistical comparisons of disease-free survival (DFS) of patients against matched, healthy control groups. This logically rigorous approach essentially equates indefinite remission with cure. The comparison

2592-434: The perspective of the patient, particularly one that has received a new treatment, the statistical model may be frustrating. It may take many years to accumulate sufficient information to determine the point at which the DFS curve flattens (and therefore no more relapses are expected). Some diseases may be discovered to be technically incurable, but also to require treatment so infrequently as to be not materially different from

2646-1100: 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

2700-431: 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

2754-540: 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

SECTION 50

#1732851384175

2808-469: 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

2862-478: 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

2916-789: Was regarded by The Lancet as the "most important advance of the [19th] Century in the medical treatment of acute infectious disease". Sulphonamides become the first widely available cure for bacterial infections. Antimalarials were first synthesized, making malaria curable. Bacterial infections became curable with the development of antibiotics. Hepatitis C , a viral infection, became curable through treatment with antiviral medications. Signs and symptoms Syndrome Disease Medical diagnosis Differential diagnosis Prognosis Acute Chronic Cure Eponymous disease Acronym or abbreviation Remission Expectation-maximization algorithm In statistics , an expectation–maximization ( EM ) algorithm

#174825