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91-411: 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 a matched control group that never had the disease. Another way of determining

182-508: A terminal patient , terminally ill or simply as being terminal . There is no standardized life expectancy for a patient to be considered terminal, although it is generally months or less. An illness which is lifelong but not fatal is called a chronic condition . Terminal patients have options for disease management after diagnosis. Examples include caregiving , continued treatment, palliative and hospice care, and physician-assisted suicide . Decisions regarding management are made by

273-408: 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

364-509: A burden on their family, and because they do not want to lose autonomy and control over their own lives among other reasons. They believe that allowing PAS is an act of compassion. While some groups believe in personal choice over death, others raise concerns regarding insurance policies and potential for abuse. According to Sulmasy et al., the major non-religious arguments against physician-assisted suicide are quoted as follows: Again, there are also arguments that there are enough protections in

455-400: 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

546-461: 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

637-420: A factor of two." There was no evidence that any specific type of clinician was better at making these predictions. Healthcare during the last year of life is costly, especially for patients who used hospital services often during end-of-life. In fact, according to Langton et al., there were "exponential increases in service use and costs as death approached." Many dying terminal patients are also brought to

728-432: 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

819-399: 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:

910-458: 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

1001-401: 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

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1092-576: A miracle cure, whether by participating in experimental treatments and clinical trials or seeking more intense treatment for the disease. Rather than to "give up fighting," patients spend thousands more dollars to try to prolong life by a few more months. What these patients often do give up, however, is quality of life at the end of life by undergoing intense and often uncomfortable treatment. A meta-analysis of 34 studies including 11,326 patients from 11 countries found that less than half of all terminal patients correctly understood their disease prognosis , or

1183-417: A patient's decision include future disability and suffering, lack of control over death, impact on family, healthcare costs, insurance coverage, personal beliefs, religious beliefs, and much more. PAS may be referred to in many different ways, such as aid in dying, assisted dying, death with dignity, and many more. These often depend on the organization and the stance they take on the issue. In this section of

1274-561: 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

1365-595: A position statement arguing against considering legalizing PAS unless comprehensive palliative care systems in the country were in place. It could be argued that with proper palliative care, the patient would experience fewer intolerable symptoms, physical or emotional, and would not choose death over these symptoms. Palliative care would also ensure that patients receive proper information about their disease prognosis as not to make decisions about PAS without complete and careful consideration. Many aspects of medical care are different for terminal patients compared to patients in

1456-867: 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

1547-399: 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

1638-547: 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 }})} ,

1729-444: A terminal stage, and they are pursuing treatment because they do not understand it to be futile . Palliative care is normally offered to terminally ill patients, regardless of their overall disease management style, if it seems likely to help manage symptoms such as pain and improve quality of life. Hospice care , which can be provided at home or in a long-term care facility, additionally provides emotional and spiritual support for

1820-486: Is a disease that cannot be cured or adequately treated and is expected to result in the death of the patient. This term is more commonly used for progressive diseases such as cancer , dementia , advanced heart disease , and for HIV/AIDS , or long COVID in bad cases, rather than for injury . In popular use, it indicates a disease that will progress until death with near absolute certainty, regardless of treatment. A patient who has such an illness may be referred to as

1911-495: 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

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2002-405: 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

2093-621: Is an attempt to improve patients' quality-of-life and comfort, and also provide support for family members and carers. Additionally, it lowers hospital admissions costs. However, needs for palliative care are often unmet whether due to lack of government support and also possible stigma associated with palliative care. For these reasons, the World Health Assembly recommends development of palliative care in health care systems. Palliative care and hospice care are often confused, and they have similar goals. However, hospice care

2184-478: 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

2275-492: 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

2366-412: Is arbitrary, and best available estimates of longevity may be incorrect. Though a given patient may properly be considered terminal, this is not a guarantee that the patient will die within six months. Similarly, a patient with a slowly progressing disease, such as AIDS , may not be considered terminally ill if the best estimate of longevity is greater than six months. However, this does not guarantee that

2457-442: Is likely to be "weeks", "months", or "years", even if more specific estimates are unavailable. However, many healthcare providers avoid telling them this because the healthcare providers are uncomfortable with death or perceive it as a professional failure. To avoid admitting that the person will inevitably die from an incurable condition, they may withhold information or, if pressed, give overly optimistic answers. For example, if

2548-689: Is limited. A common symptom that many terminal patients experience is dyspnea , or difficulty with breathing. To ease this symptom, doctors may also prescribe opioids to patients. Some studies suggest that oral opioids may help with breathlessness. However, due to lack of consistent reliable evidence, it is currently unclear whether they truly work for this purpose. Depending on the patient's condition, other medications will be prescribed accordingly. For example, if patients develop depression, antidepressants will be prescribed. Anti-inflammation and anti-nausea medications may also be prescribed. Some terminal patients opt to continue extensive treatments in hope of

2639-505: 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

2730-424: Is not disease treatment, it addresses patients' physical needs, such as pain management, offers emotional support, caring for the patient psychologically and spiritually, and helps patients build support systems that can help them get through difficult times. Palliative care can also help patients make decisions and come to understand what they want regarding their treatment goals and quality of life. Palliative care

2821-457: 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]

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2912-551: Is often incorporated into palliative care . When terminal patients are aware of their impending deaths, they have time to prepare for care, such as advance directives and living wills, which have been shown to improve end-of-life care. While death cannot be avoided, patients can strive to die a death seen as good. However, many healthcare providers are uncomfortable telling people or their families that they are dying. To avoid uncomfortable conversations, they will withhold information and evade questions. Accurately identifying

3003-488: Is often used to help providers decide and prioritize candidates for transplant. Physician-assisted suicide (PAS) is highly controversial, and legal in only a few countries. In PAS, physicians, with voluntary written and verbal consent from the patient, give patients the means to die, usually through lethal drugs. The patient then chooses to " die with dignity ," deciding on their own time and place to die. Reasons as to why patients choose PAS differ. Factors that may play into

3094-401: 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

3185-435: Is specifically for terminal patients while palliative care is more general and offered to patients who are not necessarily terminal. While hospitals focus on treating the disease, hospices focus on improving patient quality-of-life until death. Hospice patients are able to live at peace away from a hospital setting; they may live at home with a hospice provider or at an inpatient hospice facility. A common misconception

3276-439: Is that hospice care hastens death because patients "give up" fighting the disease. However, people in hospice care often live the same length of time as patients in the hospital, or longer. Additionally, people receiving hospice care have significantly lower healthcare expenditures. Hospice care allows patients to spend more time with family and friends. People in institutional (rather than home-care) hospice programs are also in

3367-616: 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

3458-447: 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

3549-469: 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

3640-762: The American College of Physicians (ACP), the American Medical Association (AMA), the World Health Organization , American Nurses Association , Hospice Nurses Association, American Psychiatric Association , and more have issued position statements against its legalization. The ACP's argument concerns the nature of the doctor-patient relationship and the tenets of the medical profession. They state that instead of using PAS to control death: "through high-quality care, effective communication, compassionate support, and

3731-524: 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

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3822-460: The emergency department (ED) at the end of life when treatment is no longer beneficial, raising costs and using limited space in the ED. While there are often claims about "disproportionate" spending of money and resources on end-of-life patients, data have not proven this type of correlation. The cost of healthcare for end-of-life patients is 13% of annual healthcare spending in the U.S. However, of

3913-443: 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

4004-404: 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

4095-494: 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

4186-517: 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

4277-506: 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

4368-475: 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

4459-527: The article, it will be referred to as PAS for the sake of consistency with the pre-existing Misplaced Pages page: Assisted Suicide . In the United States, PAS or medical aid in dying is legal in select states, including Oregon, Washington, Montana, Vermont, and New Mexico, and there are groups both in favor of and against legalization. Some groups favor PAS because they do not believe they will have control over their pain, because they believe they will be

4550-532: The caregiver works along with physicians and follows professional instructions. Caregivers may call the physician or a nurse if the individual: Most caregivers become the patient's listeners and let the individual express fears and concerns without judgment. Caregivers reassure the patient and honor all advance directives. Caregivers respect the individual's need for privacy and usually hold all information confidential. Palliative care focuses on addressing patients' needs after disease diagnosis. While palliative care

4641-443: The company of other hospice patients, which provides them with an additional support network. Terminal patients experiencing pain, especially cancer-related pain, are often prescribed opioids to relieve suffering. The specific medication prescribed, however, will differ depending on severity of pain and disease status. There exist inequities in availability of opioids to terminal patients, especially in countries where opioid access

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4732-462: 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,

4823-761: The course of their disease and likeliness of survival. This could influence patients to pursue unnecessary treatment for the disease due to unrealistic expectations. For patients with end stage kidney failure , studies have shown that transplants increase the quality of life and decreases mortality in this population. In order to be placed on the organ transplant list, patients are referred and assessed based on criteria that ranges from current comorbidities to potential for organ rejection post transplant. Initial screening measures include: blood tests, pregnancy tests, serologic tests, urinalysis, drug screening, imaging, and physical exams. For patients who are interested in liver transplantation, patients with acute liver failure have

4914-433: 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, the person is said to be cured , even though the person might someday catch another cold. Conversely,

5005-421: 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,

5096-438: 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

5187-478: 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

5278-413: 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

5369-438: 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}

5460-457: 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 ,

5551-454: The false appearance of hope . They often want to avoid the emotional outbursts that are associated with people understanding the medical situation accurately. For example, they will use death-denying language such as "She has a life-limiting diagnosis" – a term that makes the inevitable death seem less inevitable – rather than bluntly saying "No matter what we do, your daughter is almost certainly going to die from this cancer, probably within

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5642-418: 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

5733-455: The group of patients with the highest healthcare spending, end-of-life patients only made up 11% of these people, meaning the most expensive spending is not made up mostly of terminal patients. Many recent studies have shown that palliative care and hospice options as an alternative are much less expensive for end-of-life patients. Expectation-maximization algorithm In statistics , an expectation–maximization ( EM ) algorithm

5824-404: The highest priority over patients with only cirrhosis. Acute liver failure patients will present with worsening symptoms of somnolence or confusion (hepatic encephalopathy) and thinner blood (increased INR) due to the liver's inability to make clotting factors. Some patients could experience portal hypertension, hemorrhages, and abdominal swelling (ascites). Model for End Stage Liver Disease (MELD)

5915-484: The hospital for other reasons. Doctor–patient relationships are crucial in any medical setting, and especially so for terminal patients. There must be an inherent trust in the doctor to provide the best possible care for the patient. In the case of terminal illness, there is often ambiguity in communication with the patient about their condition. While terminal condition prognosis is often a grave matter, doctors do not wish to quash all hope, for it could unnecessarily harm

6006-414: The individual with daily living activities and movement. Caregivers provide assistance with food and psychological support and ensure that the individual is comfortable. The patient's family may have questions and most caregivers can provide information to help ease the mind. Doctors generally do not provide estimates for fear of instilling false hopes or obliterate an individual's hope. In most cases,

6097-485: 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

6188-459: 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} }

6279-519: The law that the slippery slope is avoided. For example, the Death with Dignity Act in Oregon includes waiting periods, multiple requests for lethal drugs, a psychiatric evaluation in the case of possible depression influencing decisions, and the patient personally swallowing the pills to ensure voluntary decision. Physicians and medical professionals also have disagreeing views on PAS. Some groups, such as

6370-457: 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

6461-485: 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

6552-834: 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

6643-435: 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

6734-690: The next few months." By definition, there is not a cure or adequate treatment for terminal illnesses. However, some kinds of medical treatments may be appropriate anyway, such as treatment to reduce pain or ease breathing. Some terminally ill patients stop all debilitating treatments to reduce unwanted side effects. Others continue aggressive treatment in the hope of an unexpected success. Still others reject conventional medical treatment and pursue unproven treatments such as radical dietary modifications. Patients' choices about different treatments may change over time. People who pursue aggressive treatment usually do not understand that their illness has reached

6825-478: 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

6916-447: 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

7007-585: 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

7098-427: The patient and loved ones. Some complementary approaches, such as relaxation therapy , massage , and acupuncture may relieve some symptoms and other causes of suffering. Terminal patients often need a caregiver , who could be a nurse , licensed practical nurse or a family member. Caregivers can help patients receive medications to reduce pain and control symptoms of nausea or vomiting . They can also assist

7189-535: The patient and their family, although medical professionals may offer recommendations of services available to terminal patients. Lifestyle after diagnosis varies depending on management decisions and the nature of the disease, and there may be restrictions depending on the condition of the patient. Terminal patients may experience depression or anxiety associated with impending death , and family and caregivers may struggle with psychological burdens. Psychotherapeutic interventions may alleviate some of these burdens, and

7280-545: The patient will not die unexpectedly early. In general, physicians slightly overestimate the survival time of terminally ill cancer patients, so that, for example, a person who is expected to live for about six weeks would likely die around four weeks. A recent systematic review on palliative patients in general, rather than specifically cancer patients, states the following: "Accuracy of categorical estimates in this systematic review ranged from 23% up to 78% and continuous estimates over-predicted actual survival by, potentially,

7371-472: The patient's mental state and have unintended consequences . However, being overly optimistic about outcomes can leave patients and families devastated when negative results arise, as is often the case with terminal illness. Often, a patient is considered terminally ill when his or her estimated life expectancy is six months or less, under the assumption that the disease will run its normal course based on previous data from other patients. The six-month standard

7462-433: 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

7553-1100: 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

7644-745: The right resources, physicians can help patients control many aspects of how they live out life's last chapter." Other groups such as the American Medical Students Association , the American Public Health Association , the American Medical Women's Association , and more support PAS as an act of compassion for the suffering patient. In many cases, the argument on PAS is also tied to proper palliative care. The International Association for Hospice and Palliative Care issued

7735-572: The start of terminal status is important because it usually occasions a review of treatment goals. Although there is no single official definition, there are four typical characteristics for determining whether a person has a terminal illness: When the remaining lifespan is expected to be days and the physical process of dying has begun, the term active dying may be used instead. Most terminally ill people are not distressed by being told that they are likely to die sooner rather than later, and they usually value knowing whether their realistic lifespan

7826-431: 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

7917-491: The typical person in that situation usually lives for two to six months, they may say only the larger number. They may rationalize the inflated claim by thinking of hopeful possibilities, such as an unproven treatment (which might shorten the person's life even further ) being attempted, or because they know that life expectancy is an imperfect estimate and could be both shorter or longer than expected. They may feel pressure from family members to give pleasant news or to preserve

8008-540: 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

8099-469: 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

8190-484: 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

8281-741: 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 Terminal illness Terminal illness or end-stage disease

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