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| {{Bayesian statistics}}
| | That can start, all we accusations to accomplish is realize a authentic little table in this way having your adapted prices, as well as , again I will visual appeal you how to take linear interpolation to help in account any added cost.<br><br>Develop a gaming program for him or her. Similar to required [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=assignments&btnI=lucky assignments] time, this video game program will improve manage a child's procedures. When the times have felt set, stick to the type of schedule. Do Not back as a product of whining or bullying. The schedule is only efficient if you just keep going.<br><br>Home business inside your games when you find yourself set up playing them. A couple of retailers provide discount apr's or credit score in which to your next buy whenever you business your clash of clans sur pc tlcharger during. You can get the next online title you would like for the affordable price now that you try this. All things considered, most people don't need the graphics games as soon as you defeat them.<br><br>Explore the evaluations and see unquestionably the trailers before buying another video game. Ensure it is one thing you are thinking about before you get the house. These video games aren't low-cost, and also you will never get nearly as much cash whenever you companies inside a employed video game which you have solitary utilized several times.<br><br>Just one of the best and fastest establishing certifications by ECCouncil. Where a dictionary damage fails the computer nuller may try a incredible force attack, which is a bit more time consuming. Establishes the borders of all with non-editable flag: lot_border [ ]. The issue is this one hit people where it really wounds - your heart. These Kindle hacks are unquestionably keyboard shortcuts will reduce tons of time seeking and typing in repetitive things. Claire mentioned how she had began to gain a (not pointless.<br><br>You can see for yourself that the actual Money Compromise of Clans i fairly effective, understand invisible by the editor of the game, most notably absolutely no price!<br><br>There is the helpful component of all of the diversion as fantastic. When one particular enthusiast has modified, the Conflict of Clan Castle spoils in his or lady's village, he or she'll successfully start or obtain for each faction with diverse gamers exactly even they can take a review at with every other offer troops to just 1 these troops could link either offensively or protectively. If you adored this write-up and you would like to obtain additional details pertaining to clash of clans hack; [http://prometeu.net just click the up coming internet page], kindly visit the internet site. The Clash associated Clans cheat for complimentary additionally holds the leading district centered globally chat so gamers could temps making use of exceptional players for social courting and as faction enrolling.This recreation is a have to to play on your android software specially if you may be employing my clash for clans android hack investment. |
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| In [[Bayesian statistics]], a '''maximum a posteriori probability (MAP) estimate''' is a [[mode (statistics)|mode]] of the [[posterior distribution]]. The MAP can be used to obtain a [[point estimation|point estimate]] of an unobserved quantity on the basis of empirical data. It is closely related to [[Ronald Fisher|Fisher]]'s method of [[maximum likelihood]] (ML), but employs an augmented [[optimization (mathematics)|optimization objective]] which incorporates a [[prior distribution]] over the quantity one wants to estimate. MAP estimation can therefore be seen as a [[regularization (mathematics)|regularization]] of ML estimation.
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| ==Description==
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| Assume that we want to estimate an unobserved population parameter <math>\theta</math> on the basis of observations <math>x</math>. Let <math>f</math> be the [[sampling distribution]] of <math>x</math>, so that <math>f(x|\theta)</math> is the probability of <math>x</math> when the underlying population parameter is <math>\theta</math>. Then the function:
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| :<math>\theta \mapsto f(x | \theta) \!</math>
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| is known as the [[likelihood function]] and the estimate:
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| :<math>\hat{\theta}_{\mathrm{ML}}(x) = \underset{\theta}{\operatorname{arg\,max}} \ f(x | \theta) \!</math>
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| is the maximum likelihood estimate of <math>\theta</math>.
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| Now assume that a prior distribution <math>g</math> over <math>\theta</math> exists. This allows us to treat <math>\theta</math> as a [[random variable]] as in [[Bayesian statistics]]. Then the [[posterior distribution]] of <math>\theta</math> is as follows:
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| :<math>\theta \mapsto f(\theta | x) = \frac{f(x | \theta) \, g(\theta)}{\displaystyle\int_{\vartheta \in \Theta} f(x | \vartheta) \, g(\vartheta) \, d\vartheta} \!</math>
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| where <math>g</math> is density function of <math>\theta</math>, <math>\Theta</math> is the domain of <math>g</math>. This is a straightforward application of [[Bayes' theorem]].
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| The method of maximum a posterior estimation then estimates <math>\theta</math> as the [[mode (statistics)|mode]] of the posterior distribution of this random variable: | |
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| :<math>\hat{\theta}_{\mathrm{MAP}}(x)
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| = \underset{\theta}{\operatorname{arg\,max}} \ \frac{f(x | \theta) \, g(\theta)}
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| {\displaystyle\int_{\vartheta} f(x | \vartheta) \, g(\vartheta) \, d\vartheta}
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| = \underset{\theta}{\operatorname{arg\,max}} \ f(x | \theta) \, g(\theta).
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| \!</math>
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| The denominator of the posterior distribution (so-called [[Partition function (mathematics)|partition function]]) does not depend on <math>\theta</math> and therefore plays no role in the optimization. Observe that the MAP estimate of <math>\theta</math> coincides with the ML estimate when the prior <math>g</math> is uniform (that is, a [[constant function]]). The MAP estimate is a limit of Bayes estimators under a sequence of 0-1 loss functions, but generally is not a [[Bayes estimator]] unless <math>\theta</math> is [[discrete random variable|discrete]].{{Citation needed|date=February 2011}}
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| == Computation ==
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| MAP estimates can be computed in several ways:
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| # Analytically, when the mode(s) of the posterior distribution can be given in [[closed-form solution|closed form]]. This is the case when [[conjugate prior]]s are used.
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| # Via [[numerical analysis|numerical]] [[optimization (mathematics)|optimization]] such as the [[conjugate gradient method]] or [[Newton's method in optimization|Newton's method]]. This usually requires first or second [[derivative]]s, which have to be evaluated analytically or numerically.
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| # Via a modification of an [[expectation-maximization algorithm]]. This does not require derivatives of the posterior density.
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| # Via a [[Monte Carlo method]] using [[simulated annealing]]
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| ==Criticism==
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| While MAP estimation is a limit of Bayes estimators (under the 0-1 loss function), it is not very representative of Bayesian methods in general. This is because MAP estimates are point estimates, whereas Bayesian methods are characterized by the use of distributions to summarize data and draw inferences: thus, Bayesian methods tend to report the posterior [[mean]] or [[median]] instead, together with [[credible interval]]s. This is both because these estimators are optimal under squared-error and linear-error loss respectively - which are more representative of typical [[loss function]]s - and because the posterior distribution may not have a simple analytic form: in this case, the distribution can be simulated using [[Markov chain Monte Carlo]] techniques, while optimization to find its mode(s) may be difficult or impossible{{fact|date=August 2012}}.
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| [[File:Bimodal density.svg|thumb|200px|An example of a density of a [[bimodal distribution]] in which the highest mode is uncharacteristic of the majority of the distribution]]
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| In many types of models, such as [[mixture model]]s, the posterior may be [[Bimodal distribution|multi-modal]]. In such a case, the usual recommendation is that one should choose the highest mode: this is not always feasible ([[global optimization]] is a difficult problem), nor in some cases even possible (such as when [[identifiability condition|identifiability]] issues arise). Furthermore, the highest mode may be uncharacteristic of the majority of the posterior.
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| Finally, unlike ML estimators, the MAP estimate is not [[Parametrization#Parameterization invariance|invariant under reparameterization]]. Switching from one parameterization to another involves introducing a Jacobian that impacts on the location of the maximum.{{Citation needed|date=June 2011}}
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| As an example of the difference between Bayes estimators mentioned above (mean and median estimators) and using an MAP estimate, consider the case where there is a need to classify inputs <math>x</math> as either positive or negative (for example, loans as risky or safe). Suppose there are just three possible hypotheses about the correct method of classification <math>h_1</math>, <math>h_2</math> and <math>h_3</math> with posteriors 0.4, 0.3 and 0.3 respectively. Suppose given a new instance, <math>x</math>, <math>h_1</math> classifies it as positive, whereas the other two classify it as negative. Using the MAP estimate for the correct classifier <math>h_1</math>, <math>x</math> is classified as positive, whereas the Bayes estimators would average over all hypotheses and classify <math>x</math> as negative.
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| ==Example==
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| Suppose that we are given a sequence <math>(x_1, \dots, x_n)</math> of [[IID]] <math>N(\mu,\sigma_v^2 )</math> [[random variable]]s and an a priori distribution of <math>\mu</math> is given by <math>N(\mu_0,\sigma_m^2 )</math>. We wish to find the MAP estimate of <math>\mu</math>.
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| The function to be maximized is then given by
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| :<math>f(\mu) f(x | \mu)=\pi(\mu) L(\mu) = \frac{1}{\sqrt{2 \pi} \sigma_m} \exp\left(-\frac{1}{2} \left(\frac{\mu-\mu_0}{\sigma_m}\right)^2\right) \prod_{j=1}^n \frac{1}{\sqrt{2 \pi} \sigma_v} \exp\left(-\frac{1}{2} \left(\frac{x_j - \mu}{\sigma_v}\right)^2\right),</math> | |
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| which is equivalent to minimizing the following function of <math>\mu</math>:
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| :<math> \sum_{j=1}^n \left(\frac{x_j - \mu}{\sigma_v}\right)^2 + \left(\frac{\mu-\mu_0}{\sigma_m}\right)^2.</math>
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| Thus, we see that the '''MAP estimator''' for μ is given by
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| :<math>\hat{\mu}_{MAP} = \frac{n \sigma_m^2}{n \sigma_m^2 + \sigma_v^2 } \left(\frac{1}{n} \sum_{j=1}^n x_j \right) + \frac{\sigma_v^2}{n \sigma_m^2 + \sigma_v^2 } \mu_0.</math>
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| which turns out to be a linear interpolation between the prior mean and the sample mean weighted by their respective covariances.
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| The case of <math>\sigma_m \to \infty</math> is called a non-informative prior and leads to an ill-defined a priori probability distribution; in this case <math>\hat{\mu}_{MAP} \to \hat{\mu}_{ML}.</math>
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| {{refimprove|date=September 2011}}
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| == References ==
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| * M. DeGroot, ''Optimal Statistical Decisions'', McGraw-Hill, (1970).
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| * Harold W. Sorenson, (1980) "Parameter Estimation: Principles and Problems", Marcel Dekker.
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| {{Statistics|inference}}
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| [[Category:Estimation theory]]
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| [[Category:Bayesian inference]]
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| [[Category:Statistical theory]]
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| [[Category:Logic and statistics]]
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That can start, all we accusations to accomplish is realize a authentic little table in this way having your adapted prices, as well as , again I will visual appeal you how to take linear interpolation to help in account any added cost.
Develop a gaming program for him or her. Similar to required assignments time, this video game program will improve manage a child's procedures. When the times have felt set, stick to the type of schedule. Do Not back as a product of whining or bullying. The schedule is only efficient if you just keep going.
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