Misplaced Pages

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116-510: In maximum likelihood estimation , the argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often approximated by the likelihood's Hessian matrix at the maximum) gives an indication of the estimate's precision . In contrast, in Bayesian statistics , the estimate of interest is the converse of the likelihood,

232-492: A r g m a x θ ⁡ 1 h L ( θ ∣ x ∈ [ x j , x j + h ] ) , {\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h]),} since h {\textstyle h}

348-661: A r g m a x θ ⁡ 1 h ∫ x j x j + h f ( x ∣ θ ) d x , {\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\Pr(x_{j}\leq x\leq x_{j}+h\mid \theta )=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx,} where f ( x ∣ θ ) {\textstyle f(x\mid \theta )}

464-420: A r g m a x θ ⁡ L ( θ ∣ x j ) = a r g m a x θ ⁡ [ lim h → 0 + L ( θ ∣ x ∈ [ x j , x j + h ] ) ] =

580-471: A r g m a x θ ⁡ L ( θ ∣ x j ) = a r g m a x θ ⁡ f ( x j ∣ θ ) , {\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x_{j})=\mathop {\operatorname {arg\,max} } _{\theta }f(x_{j}\mid \theta ),} and so maximizing

696-933: A r g m a x θ ⁡ [ lim h → 0 + 1 h ∫ x j x j + h f ( x ∣ θ ) d x ] = a r g m a x θ ⁡ f ( x j ∣ θ ) . {\displaystyle {\begin{aligned}&\mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x_{j})=\mathop {\operatorname {arg\,max} } _{\theta }\left[\lim _{h\to 0^{+}}{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])\right]\\[4pt]={}&\mathop {\operatorname {arg\,max} } _{\theta }\left[\lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx\right]=\mathop {\operatorname {arg\,max} } _{\theta }f(x_{j}\mid \theta ).\end{aligned}}} Therefore,

812-575: A log-normal distribution . The density of Y follows with f X {\displaystyle f_{X}} standard Normal and g − 1 ( y ) = log ⁡ ( y ) {\displaystyle g^{-1}(y)=\log(y)} , | ( g − 1 ( y ) ) ′ | = 1 y {\displaystyle |(g^{-1}(y))^{\prime }|={\frac {1}{y}}} for y > 0 {\displaystyle y>0} . As assumed above, if

928-401: A parametric family { f ( ⋅ ; θ ) ∣ θ ∈ Θ } , {\displaystyle \;\{f(\cdot \,;\theta )\mid \theta \in \Theta \}\;,} where Θ {\displaystyle \,\Theta \,} is called the parameter space , a finite-dimensional subset of Euclidean space . Evaluating

1044-430: A vector-valued function mapping R k {\displaystyle \,\mathbb {R} ^{k}\,} into R r   . {\displaystyle \;\mathbb {R} ^{r}~.} Estimating the true parameter θ {\displaystyle \theta } belonging to Θ {\displaystyle \Theta } then, as a practical matter, means to find

1160-482: A classifier that minimizes total expected risk, especially, when the costs (the loss function) associated with different decisions are equal, the classifier is minimizing the error over the whole distribution. Thus, the Bayes Decision Rule is stated as where w 1 , w 2 {\displaystyle \;w_{1}\,,w_{2}\;} are predictions of different classes. From

1276-626: A combination of mathematical and philosophical support for frequentism in the era. According to the Oxford English Dictionary , the term frequentist was first used by M.G. Kendall in 1949, to contrast with Bayesians , whom he called non-frequentists . Kendall observed "The Frequency Theory of Probability" was used a generation earlier as a chapter title in Keynes (1921). The historical sequence: The primary historical sources in probability and statistics did not use

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1392-447: A conclusion which could only be reached via Bayes' theorem given knowledge about the marginal probabilities P ( p H = 0.5 ) {\textstyle P(p_{\text{H}}=0.5)} and P ( HH ) {\textstyle P({\text{HH}})} . Now suppose that the coin is not a fair coin, but instead that p H = 0.3 {\textstyle p_{\text{H}}=0.3} . Then

1508-407: A density f ( x ∣ θ ) {\textstyle f(x\mid \theta )} , where the sum of all the p {\textstyle p} 's added to the integral of f {\textstyle f} is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to

1624-529: A factor that does not depend on the model parameters. For example, the MLE parameters of the log-normal distribution are the same as those of the normal distribution fitted to the logarithm of the data. In fact, in the log-normal case if X ∼ N ( 0 , 1 ) {\displaystyle X\sim {\mathcal {N}}(0,1)} , then Y = g ( X ) = e X {\displaystyle Y=g(X)=e^{X}} follows

1740-487: A given significance level . Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof. The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by Wilks' theorem . The likelihood ratio is also of central importance in Bayesian inference , where it is known as the Bayes factor , and is used in Bayes' rule . Stated in terms of odds , Bayes' rule states that

1856-437: A number of repetitions of the experiment, is a measure of the probability of that event. This is the core conception of probability in the frequentist interpretation. A claim of the frequentist approach is that, as the number of trials increases, the change in the relative frequency will diminish. Hence, one can view a probability as the limiting value of the corresponding relative frequencies. The frequentist interpretation

1972-464: A parameter θ {\textstyle \theta } . Then the function L ( θ ∣ x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta \mid x)=f_{\theta }(x),} considered as a function of θ {\textstyle \theta } , is the likelihood function (of θ {\textstyle \theta } , given

2088-440: A parameter θ {\textstyle \theta } . Then the function L ( θ ∣ x ) = p θ ( x ) = P θ ( X = x ) , {\displaystyle {\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x),} considered as a function of θ {\textstyle \theta } ,

2204-573: A perspective of minimizing error, it can also be stated as where Frequentist probability Frequentist probability or frequentism is an interpretation of probability ; it defines an event's probability as the limit of its relative frequency in infinitely many trials (the long-run probability ). Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question. The development of

2320-741: A probability density or mass function x ↦ f ( x ∣ θ ) , {\displaystyle x\mapsto f(x\mid \theta ),} where x {\textstyle x} is a realization of the random variable X {\textstyle X} , the likelihood function is θ ↦ f ( x ∣ θ ) , {\displaystyle \theta \mapsto f(x\mid \theta ),} often written L ( θ ∣ x ) . {\displaystyle {\mathcal {L}}(\theta \mid x).} In other words, when f ( x ∣ θ ) {\textstyle f(x\mid \theta )}

2436-494: A set h 1 , h 2 , … , h r , h r + 1 , … , h k {\displaystyle \;h_{1},h_{2},\ldots ,h_{r},h_{r+1},\ldots ,h_{k}\;} in such a way that h ∗ = [ h 1 , h 2 , … , h k ] {\displaystyle \;h^{\ast }=\left[h_{1},h_{2},\ldots ,h_{k}\right]\;}

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2552-535: A statistical test of the "validity" of the constraint, known as the Lagrange multiplier test . Nonparametric maximum likelihood estimation can be performed using the empirical likelihood . A maximum likelihood estimator is an extremum estimator obtained by maximizing, as a function of θ , the objective function ℓ ^ ( θ ; x ) {\displaystyle {\widehat {\ell \,}}(\theta \,;x)} . If

2668-439: A stronger condition of uniform convergence almost surely has to be imposed: Additionally, if (as assumed above) the data were generated by f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})} , then under certain conditions, it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution. Specifically, where I

2784-418: A unique global maximum. Compactness implies that the likelihood cannot approach the maximum value arbitrarily close at some other point (as demonstrated for example in the picture on the right). Compactness is only a sufficient condition and not a necessary condition. Compactness can be replaced by some other conditions, such as: The dominance condition can be employed in the case of i.i.d. observations. In

2900-500: Is negative definite for every θ ∈ Θ {\textstyle \,\theta \in \Theta \,} at which the gradient ∇ L ≡ [ ∂ L ∂ θ i ] i = 1 n i {\textstyle \;\nabla L\equiv \left[\,{\frac {\partial L}{\,\partial \theta _{i}\,}}\,\right]_{i=1}^{n_{\mathrm {i} }}\;} vanishes, and if

3016-500: Is negative semi-definite at θ ^ {\displaystyle {\widehat {\theta \,}}} , as this indicates local concavity . Conveniently, most common probability distributions – in particular the exponential family – are logarithmically concave . While the domain of the likelihood function—the parameter space —is generally a finite-dimensional subset of Euclidean space , additional restrictions sometimes need to be incorporated into

3132-635: Is positive definite and | I ( θ ) | {\textstyle \,\left|\mathbf {I} (\theta )\right|\,} is finite. This ensures that the score has a finite variance. The above conditions are sufficient, but not necessary. That is, a model that does not meet these regularity conditions may or may not have a maximum likelihood estimator of the properties mentioned above. Further, in case of non-independently or non-identically distributed observations additional properties may need to be assumed. In Bayesian statistics, almost identical regularity conditions are imposed on

3248-472: Is a one-to-one function from R k {\displaystyle \mathbb {R} ^{k}} to itself, and reparameterize the likelihood function by setting ϕ i = h i ( θ 1 , θ 2 , … , θ k )   . {\displaystyle \;\phi _{i}=h_{i}(\theta _{1},\theta _{2},\ldots ,\theta _{k})~.} Because of

3364-499: Is a column-vector of Lagrange multipliers and ∂ h ( θ ) T ∂ θ {\displaystyle \;{\frac {\partial h(\theta )^{\mathsf {T}}}{\partial \theta }}\;} is the k × r Jacobian matrix of partial derivatives. Naturally, if the constraints are not binding at the maximum, the Lagrange multipliers should be zero. This in turn allows for

3480-407: Is a method of estimating the parameters of an assumed probability distribution , given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model , the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood

3596-471: Is a model, often in idealized form, of the process generated by the data. It is a common aphorism in statistics that all models are wrong . Thus, true consistency does not occur in practical applications. Nevertheless, consistency is often considered to be a desirable property for an estimator to have. To establish consistency, the following conditions are sufficient. In other words, different parameter values θ correspond to different distributions within

Likelihood function - Misplaced Pages Continue

3712-465: Is a philosophical approach to the definition and use of probabilities; it is one of several such approaches. It does not claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages. As an interpretation, it is not in conflict with the mathematical axiomatization of probability theory; rather, it provides guidance for how to apply mathematical probability theory to real-world situations. It offers distinct guidance in

3828-625: Is a real upper triangular matrix and Γ T {\displaystyle \Gamma ^{\mathsf {T}}} is its transpose . In practice, restrictions are usually imposed using the method of Lagrange which, given the constraints as defined above, leads to the restricted likelihood equations where   λ = [ λ 1 , λ 2 , … , λ r ] T   {\displaystyle ~\lambda =\left[\lambda _{1},\lambda _{2},\ldots ,\lambda _{r}\right]^{\mathsf {T}}~}

3944-570: Is assumed that the information matrix , I ( θ ) = ∫ − ∞ ∞ ∂ log ⁡ f ∂ θ r   ∂ log ⁡ f ∂ θ s   f   d z {\displaystyle \mathbf {I} (\theta )=\int _{-\infty }^{\infty }{\frac {\partial \log f}{\partial \theta _{r}}}\ {\frac {\partial \log f}{\partial \theta _{s}}}\ f\ \mathrm {d} z}

4060-412: Is both intuitive and flexible, and as such the method has become a dominant means of statistical inference . If the likelihood function is differentiable , the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes

4176-445: Is called the maximum likelihood estimator . It is generally a function defined over the sample space , i.e. taking a given sample as its argument. A sufficient but not necessary condition for its existence is for the likelihood function to be continuous over a parameter space Θ {\displaystyle \,\Theta \,} that is compact . For an open Θ {\displaystyle \,\Theta \,}

4292-494: Is central to likelihoodist statistics : the law of likelihood states that degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio. In frequentist inference , the likelihood ratio is the basis for a test statistic , the so-called likelihood-ratio test . By the Neyman–Pearson lemma , this is the most powerful test for comparing two simple hypotheses at

4408-512: Is defined to be { θ : R ( θ ) ≥ p 100 } . {\displaystyle \left\{\theta :R(\theta )\geq {\frac {p}{100}}\right\}.} If θ is a single real parameter, a p % likelihood region will usually comprise an interval of real values. If the region does comprise an interval, then it is called a likelihood interval . Maximum likelihood estimation In statistics , maximum likelihood estimation ( MLE )

4524-411: Is defined to be R ( θ ) = L ( θ ∣ x ) L ( θ ^ ∣ x ) . {\displaystyle R(\theta )={\frac {{\mathcal {L}}(\theta \mid x)}{{\mathcal {L}}({\hat {\theta }}\mid x)}}.} Thus, the relative likelihood is the likelihood ratio (discussed above) with

4640-452: Is given by L ( θ ∣ x ∈ [ x j , x j + h ] ) {\textstyle {\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])} . Observe that a r g m a x θ ⁡ L ( θ ∣ x ∈ [ x j , x j + h ] ) =

4756-460: Is not directly used in AIC-based statistics. Instead, what is used is the relative likelihood of models (see below). In evidence-based medicine , likelihood ratios are used in diagnostic testing to assess the value of performing a diagnostic test . Since the actual value of the likelihood function depends on the sample, it is often convenient to work with a standardized measure. Suppose that

Likelihood function - Misplaced Pages Continue

4872-433: Is often avoided and instead f ( x ; θ ) {\textstyle f(x;\theta )} or f ( x , θ ) {\textstyle f(x,\theta )} are used to indicate that θ {\textstyle \theta } is regarded as a fixed unknown quantity rather than as a random variable being conditioned on. The likelihood function does not specify

4988-481: Is positive and constant. Because a r g m a x θ ⁡ 1 h L ( θ ∣ x ∈ [ x j , x j + h ] ) = a r g m a x θ ⁡ 1 h Pr ( x j ≤ x ≤ x j + h ∣ θ ) =

5104-508: Is possible to estimate the second-order bias of the maximum likelihood estimator, and correct for that bias by subtracting it: This estimator is unbiased up to the terms of order ⁠ 1 /   n   ⁠ , and is called the bias-corrected maximum likelihood estimator . This bias-corrected estimator is second-order efficient (at least within the curved exponential family), meaning that it has minimal mean squared error among all second-order bias-corrected estimators, up to

5220-406: Is such that ∫ − ∞ ∞ H r s t ( z ) d z ≤ M < ∞ . {\textstyle \,\int _{-\infty }^{\infty }H_{rst}(z)\mathrm {d} z\leq M<\infty \;.} This boundedness of the derivatives is needed to allow for differentiation under the integral sign . And lastly, it

5336-498: Is taken with respect to the true density. Maximum-likelihood estimators have no optimum properties for finite samples, in the sense that (when evaluated on finite samples) other estimators may have greater concentration around the true parameter-value. However, like other estimation methods, maximum likelihood estimation possesses a number of attractive limiting properties : As the sample size increases to infinity, sequences of maximum likelihood estimators have these properties: Under

5452-403: Is that in finite samples, there may exist multiple roots for the likelihood equations. Whether the identified root θ ^ {\displaystyle \,{\widehat {\theta \,}}\,} of the likelihood equations is indeed a (local) maximum depends on whether the matrix of second-order partial and cross-partial derivatives, the so-called Hessian matrix

5568-517: Is the Fisher information matrix . The maximum likelihood estimator selects the parameter value which gives the observed data the largest possible probability (or probability density, in the continuous case). If the parameter consists of a number of components, then we define their separate maximum likelihood estimators, as the corresponding component of the MLE of the complete parameter. Consistent with this, if θ ^ {\displaystyle {\widehat {\theta \,}}}

5684-456: Is the likelihood function , given the outcome x {\textstyle x} of the random variable X {\textstyle X} . Sometimes the probability of "the value x {\textstyle x} of X {\textstyle X} for the parameter value θ {\textstyle \theta }   " is written as P ( X = x | θ ) or P ( X = x ; θ ) . The likelihood

5800-400: Is the MLE for θ {\displaystyle \theta } , and if g ( θ ) {\displaystyle g(\theta )} is any transformation of θ {\displaystyle \theta } , then the MLE for α = g ( θ ) {\displaystyle \alpha =g(\theta )} is by definition It maximizes

5916-422: Is the index of the discrete probability mass corresponding to observation x {\textstyle x} , because maximizing the probability mass (or probability) at x {\textstyle x} amounts to maximizing the likelihood of the specific observation. The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and

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6032-505: Is the posterior probability of θ {\textstyle \theta } given the data x {\textstyle x} . Consider a simple statistical model of a coin flip: a single parameter p H {\textstyle p_{\text{H}}} that expresses the "fairness" of the coin. The parameter is the probability that a coin lands heads up ("H") when tossed. p H {\textstyle p_{\text{H}}} can take on any value within

6148-1196: Is the probability density function, it follows that a r g m a x θ ⁡ L ( θ ∣ x ∈ [ x j , x j + h ] ) = a r g m a x θ ⁡ 1 h ∫ x j x j + h f ( x ∣ θ ) d x . {\displaystyle \mathop {\operatorname {arg\,max} } _{\theta }{\mathcal {L}}(\theta \mid x\in [x_{j},x_{j}+h])=\mathop {\operatorname {arg\,max} } _{\theta }{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx.} The first fundamental theorem of calculus provides that lim h → 0 + 1 h ∫ x j x j + h f ( x ∣ θ ) d x = f ( x j ∣ θ ) . {\displaystyle \lim _{h\to 0^{+}}{\frac {1}{h}}\int _{x_{j}}^{x_{j}+h}f(x\mid \theta )\,dx=f(x_{j}\mid \theta ).} Then

6264-547: Is the probability of the data averaged over all parameters. Since the denominator is independent of θ , the Bayesian estimator is obtained by maximizing f ( x 1 , x 2 , … , x n ∣ θ ) P ⁡ ( θ ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )\operatorname {\mathbb {P} } (\theta )} with respect to θ . If we further assume that

6380-661: Is the probability that a particular outcome x {\textstyle x} is observed when the true value of the parameter is θ {\textstyle \theta } , equivalent to the probability mass on x {\textstyle x} ; it is not a probability density over the parameter θ {\textstyle \theta } . The likelihood, L ( θ ∣ x ) {\textstyle {\mathcal {L}}(\theta \mid x)} , should not be confused with P ( θ ∣ x ) {\textstyle P(\theta \mid x)} , which

6496-439: Is this density interpreted as a function of the parameter, rather than the random variable. Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are Radon–Nikodym derivatives with respect to the same dominating measure.) The above discussion of the likelihood for discrete random variables uses

6612-485: Is viewed as a function of x {\textstyle x} with θ {\textstyle \theta } fixed, it is a probability density function, and when viewed as a function of θ {\textstyle \theta } with x {\textstyle x} fixed, it is a likelihood function. In the frequentist paradigm , the notation f ( x ∣ θ ) {\textstyle f(x\mid \theta )}

6728-453: The Cramér–Rao bound . Specifically, where   I   {\displaystyle ~{\mathcal {I}}~} is the Fisher information matrix : In particular, it means that the bias of the maximum likelihood estimator is equal to zero up to the order ⁠ 1 / √ n   ⁠ . However, when we consider the higher-order terms in the expansion of

6844-431: The counting measure , under which the probability density at any outcome equals the probability of that outcome. The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses p k ( θ ) {\textstyle p_{k}(\theta )} and

6960-523: The matrix of second partials H ( θ ) ≡ [ ∂ 2 L ∂ θ i ∂ θ j ] i , j = 1 , 1 n i , n j {\displaystyle \mathbf {H} (\theta )\equiv \left[\,{\frac {\partial ^{2}L}{\,\partial \theta _{i}\,\partial \theta _{j}\,}}\,\right]_{i,j=1,1}^{n_{\mathrm {i} },n_{\mathrm {j} }}\;}

7076-534: The maximum a posteriori estimate is the parameter θ that maximizes the probability of θ given the data, given by Bayes' theorem: where P ⁡ ( θ ) {\displaystyle \operatorname {\mathbb {P} } (\theta )} is the prior distribution for the parameter θ and where P ⁡ ( x 1 , x 2 , … , x n ) {\displaystyle \operatorname {\mathbb {P} } (x_{1},x_{2},\ldots ,x_{n})}

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7192-407: The maximum likelihood estimate for the parameter θ is θ ^ {\textstyle {\hat {\theta }}} . Relative plausibilities of other θ values may be found by comparing the likelihoods of those other values with the likelihood of θ ^ {\textstyle {\hat {\theta }}} . The relative likelihood of θ

7308-426: The outcome X = x {\textstyle X=x} ). Again, L {\textstyle {\mathcal {L}}} is not a probability density or mass function over θ {\textstyle \theta } , despite being a function of θ {\textstyle \theta } given the observation X = x {\textstyle X=x} . The use of

7424-713: The posterior odds of two alternatives, ⁠ A 1 {\displaystyle A_{1}} ⁠ and ⁠ A 2 {\displaystyle A_{2}} ⁠ , given an event ⁠ B {\displaystyle B} ⁠ , is the prior odds, times the likelihood ratio. As an equation: O ( A 1 : A 2 ∣ B ) = O ( A 1 : A 2 ) ⋅ Λ ( A 1 : A 2 ∣ B ) . {\displaystyle O(A_{1}:A_{2}\mid B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\mid B).} The likelihood ratio

7540-404: The probability density in specifying the likelihood function above is justified as follows. Given an observation x j {\textstyle x_{j}} , the likelihood for the interval [ x j , x j + h ] {\textstyle [x_{j},x_{j}+h]} , where h > 0 {\textstyle h>0} is a constant,

7656-414: The Bayesian estimator coincides with the maximum likelihood estimator for a uniform prior distribution P ⁡ ( θ ) {\displaystyle \operatorname {\mathbb {P} } (\theta )} . In many practical applications in machine learning , maximum-likelihood estimation is used as the model for parameter estimation. The Bayesian Decision theory is about designing

7772-523: The best interpretation of probability available to them was frequentist. All were suspicious of "inverse probability" (the available alternative) with prior probabilities chosen by using the principle of indifference. Fisher said, "... the theory of inverse probability is founded upon an error, [referring to Bayes theorem] and must be wholly rejected." While Neyman was a pure frequentist, Fisher's views of probability were unique: Both Fisher and Neyman had nuanced view of probability. von Mises offered

7888-423: The concept of frequentist probability and published a critical proof (the weak law of large numbers ) posthumously (Bernoulli, 1713). He is also credited with some appreciation for subjective probability (prior to and without Bayes theorem ). Gauss and Laplace used frequentist (and other) probability in derivations of the least squares method a century later, a generation before Poisson. Laplace considered

8004-460: The conditions outlined below, the maximum likelihood estimator is consistent . The consistency means that if the data were generated by f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})} and we have a sufficiently large number of observations n , then it is possible to find the value of θ 0 with arbitrary precision. In mathematical terms this means that as n goes to infinity

8120-415: The construction and design of practical experiments, especially when contrasted with the Bayesian interpretation . As to whether this guidance is useful, or is apt to mis-interpretation, has been a source of controversy. Particularly when the frequency interpretation of probability is mistakenly assumed to be the only possible basis for frequentist inference . So, for example, a list of mis-interpretations of

8236-605: The corresponding likelihood. The result of such calculations is displayed in Figure ;1. The integral of L {\textstyle {\mathcal {L}}} over [0, 1] is 1/3; likelihoods need not integrate or sum to one over the parameter space. Let X {\textstyle X} be a random variable following an absolutely continuous probability distribution with density function f {\textstyle f} (a function of x {\textstyle x} ) which depends on

8352-534: The current terminology of classical , subjective (Bayesian), and frequentist probability. Probability theory is a branch of mathematics. While its roots reach centuries into the past, it reached maturity with the axioms of Andrey Kolmogorov in 1933. The theory focuses on the valid operations on probability values rather than on the initial assignment of values; the mathematics is largely independent of any interpretation of probability. Applications and interpretations of probability are considered by philosophy,

8468-430: The data are independent and identically distributed , then we have this being the sample analogue of the expected log-likelihood ℓ ( θ ) = E ⁡ [ ln ⁡ f ( x i ∣ θ ) ] {\displaystyle \ell (\theta )=\operatorname {\mathbb {E} } [\,\ln f(x_{i}\mid \theta )\,]} , where this expectation

8584-414: The data were generated by   f ( ⋅ ; θ 0 )   , {\displaystyle ~f(\cdot \,;\theta _{0})~,} then under certain conditions, it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution. It is √ n   -consistent and asymptotically efficient, meaning that it reaches

8700-499: The density component, the likelihood function for an observation from the continuous component can be dealt with in the manner shown above. For an observation from the discrete component, the likelihood function for an observation from the discrete component is simply L ( θ ∣ x ) = p k ( θ ) , {\displaystyle {\mathcal {L}}(\theta \mid x)=p_{k}(\theta ),} where k {\textstyle k}

8816-464: The distribution of this estimator, it turns out that θ mle has bias of order 1 ⁄ n . This bias is equal to (componentwise) where I j k {\displaystyle {\mathcal {I}}^{jk}} (with superscripts) denotes the ( j,k )-th component of the inverse Fisher information matrix I − 1 {\displaystyle {\mathcal {I}}^{-1}} , and Using these formulae it

8932-450: The early 20th century included Fisher , Neyman , and Pearson . Fisher contributed to most of statistics and made significance testing the core of experimental science, although he was critical of the frequentist concept of "repeated sampling from the same population" ; Neyman formulated confidence intervals and contributed heavily to sampling theory; Neyman and Pearson paired in the creation of hypothesis testing. All valued objectivity, so

9048-549: The equivariance of the maximum likelihood estimator, the properties of the MLE apply to the restricted estimates also. For instance, in a multivariate normal distribution the covariance matrix Σ {\displaystyle \,\Sigma \,} must be positive-definite ; this restriction can be imposed by replacing Σ = Γ T Γ , {\displaystyle \;\Sigma =\Gamma ^{\mathsf {T}}\Gamma \;,} where Γ {\displaystyle \Gamma }

9164-400: The estimation process. The parameter space can be expressed as where h ( θ ) = [ h 1 ( θ ) , h 2 ( θ ) , … , h r ( θ ) ] {\displaystyle \;h(\theta )=\left[h_{1}(\theta ),h_{2}(\theta ),\ldots ,h_{r}(\theta )\right]\;} is

9280-565: The estimator θ ^ {\displaystyle {\widehat {\theta \,}}} converges in probability to its true value: Under slightly stronger conditions, the estimator converges almost surely (or strongly ): In practical applications, data is never generated by f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})} . Rather, f ( ⋅ ; θ 0 ) {\displaystyle f(\cdot \,;\theta _{0})}

9396-1178: The existence of a Taylor expansion . Second, for almost all x {\textstyle x} and for every θ ∈ Θ {\textstyle \,\theta \in \Theta \,} it must be that | ∂ f ∂ θ r | < F r ( x ) , | ∂ 2 f ∂ θ r ∂ θ s | < F r s ( x ) , | ∂ 3 f ∂ θ r ∂ θ s ∂ θ t | < H r s t ( x ) {\displaystyle \left|{\frac {\partial f}{\partial \theta _{r}}}\right|<F_{r}(x)\,,\quad \left|{\frac {\partial ^{2}f}{\partial \theta _{r}\,\partial \theta _{s}}}\right|<F_{rs}(x)\,,\quad \left|{\frac {\partial ^{3}f}{\partial \theta _{r}\,\partial \theta _{s}\,\partial \theta _{t}}}\right|<H_{rst}(x)} where H {\textstyle H}

9512-461: The existence of a global maximum of the likelihood function is of the utmost importance. By the extreme value theorem , it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist. While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values might be unknown. In that case, concavity of

9628-413: The fixed denominator L ( θ ^ ) {\textstyle {\mathcal {L}}({\hat {\theta }})} . This corresponds to standardizing the likelihood to have a maximum of 1. A likelihood region is the set of all values of θ whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a p % likelihood region for θ

9744-505: The frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation . In the classical interpretation, probability was defined in terms of the principle of indifference , based on the natural symmetry of a problem, so, for example, the probabilities of dice games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry for reasoning. In

9860-447: The frequentist interpretation, probabilities are discussed only when dealing with well-defined random experiments. The set of all possible outcomes of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space to be considered. For any given event, only one of two possibilities may hold: It occurs or it does not. The relative frequency of occurrence of an event, observed in

9976-508: The joint density at the observed data sample y = ( y 1 , y 2 , … , y n ) {\displaystyle \;\mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})\;} gives a real-valued function, which is called the likelihood function . For independent and identically distributed random variables , f n ( y ; θ ) {\displaystyle f_{n}(\mathbf {y} ;\theta )} will be

10092-419: The likelihood function L n {\displaystyle \,{\mathcal {L}}_{n}\,} is called the maximum likelihood estimate. Further, if the function θ ^ n : R n → Θ {\displaystyle \;{\hat {\theta }}_{n}:\mathbb {R} ^{n}\to \Theta \;} so defined is measurable , then it

10208-405: The likelihood function approaches a constant on the boundary of the parameter space, ∂ Θ , {\textstyle \;\partial \Theta \;,} i.e., lim θ → ∂ Θ L ( θ ) = 0 , {\displaystyle \lim _{\theta \to \partial \Theta }L(\theta )=0\;,} which may include

10324-712: The likelihood function in order to proof asymptotic normality of the posterior probability , and therefore to justify a Laplace approximation of the posterior in large samples. A likelihood ratio is the ratio of any two specified likelihoods, frequently written as: Λ ( θ 1 : θ 2 ∣ x ) = L ( θ 1 ∣ x ) L ( θ 2 ∣ x ) . {\displaystyle \Lambda (\theta _{1}:\theta _{2}\mid x)={\frac {{\mathcal {L}}(\theta _{1}\mid x)}{{\mathcal {L}}(\theta _{2}\mid x)}}.} The likelihood ratio

10440-411: The likelihood function may increase without ever reaching a supremum value. In practice, it is often convenient to work with the natural logarithm of the likelihood function, called the log-likelihood : Since the logarithm is a monotonic function , the maximum of ℓ ( θ ; y ) {\displaystyle \;\ell (\theta \,;\mathbf {y} )\;} occurs at

10556-516: The likelihood function plays a key role. More specifically, if the likelihood function is twice continuously differentiable on the k -dimensional parameter space Θ {\textstyle \Theta } assumed to be an open connected subset of R k , {\textstyle \mathbb {R} ^{k}\,,} there exists a unique maximum θ ^ ∈ Θ {\textstyle {\hat {\theta }}\in \Theta } if

10672-501: The likelihood of observing "HH" assuming p H = 0.5 {\textstyle p_{\text{H}}=0.5} is L ( p H = 0.5 ∣ HH ) = 0.25. {\displaystyle {\mathcal {L}}(p_{\text{H}}=0.5\mid {\text{HH}})=0.25.} This is not the same as saying that P ( p H = 0.5 ∣ H H ) = 0.25 {\textstyle P(p_{\text{H}}=0.5\mid HH)=0.25} ,

10788-409: The likelihood when the random errors are assumed to have normal distributions with the same variance. From the perspective of Bayesian inference , MLE is generally equivalent to maximum a posteriori (MAP) estimation with a prior distribution that is uniform in the region of interest. In frequentist inference , MLE is a special case of an extremum estimator , with the objective function being

10904-714: The likelihood. We model a set of observations as a random sample from an unknown joint probability distribution which is expressed in terms of a set of parameters . The goal of maximum likelihood estimation is to determine the parameters for which the observed data have the highest joint probability. We write the parameters governing the joint distribution as a vector θ = [ θ 1 , θ 2 , … , θ k ] T {\displaystyle \;\theta =\left[\theta _{1},\,\theta _{2},\,\ldots ,\,\theta _{k}\right]^{\mathsf {T}}\;} so that this distribution falls within

11020-498: The maximum of the likelihood function subject to the constraint   h ( θ ) = 0   . {\displaystyle ~h(\theta )=0~.} Theoretically, the most natural approach to this constrained optimization problem is the method of substitution, that is "filling out" the restrictions h 1 , h 2 , … , h r {\displaystyle \;h_{1},h_{2},\ldots ,h_{r}\;} to

11136-410: The meaning of p-values accompanies the article on p -values; controversies are detailed in the article on statistical hypothesis testing . The Jeffreys–Lindley paradox shows how different interpretations, applied to the same data set, can lead to different conclusions about the 'statistical significance' of a result. As Feller notes: There is no place in our system for speculations concerning

11252-412: The model. If this condition did not hold, there would be some value θ 1 such that θ 0 and θ 1 generate an identical distribution of the observable data. Then we would not be able to distinguish between these two parameters even with an infinite amount of data—these parameters would have been observationally equivalent . The identification condition establishes that the log-likelihood has

11368-547: The non-i.i.d. case, the uniform convergence in probability can be checked by showing that the sequence ℓ ^ ( θ ∣ x ) {\displaystyle {\widehat {\ell \,}}(\theta \mid x)} is stochastically equicontinuous . If one wants to demonstrate that the ML estimator θ ^ {\displaystyle {\widehat {\theta \,}}} converges to θ 0 almost surely , then

11484-430: The occurrence of a maximum (or a minimum) are known as the likelihood equations. For some models, these equations can be explicitly solved for θ ^ , {\displaystyle \,{\widehat {\theta \,}}\,,} but in general no closed-form solution to the maximization problem is known or available, and an MLE can only be found via numerical optimization . Another problem

11600-426: The points at infinity if Θ {\textstyle \,\Theta \,} is unbounded. Mäkeläinen and co-authors prove this result using Morse theory while informally appealing to a mountain pass property. Mascarenhas restates their proof using the mountain pass theorem . In the proofs of consistency and asymptotic normality of the maximum likelihood estimator, additional assumptions are made about

11716-445: The prior P ⁡ ( θ ) {\displaystyle \operatorname {\mathbb {P} } (\theta )} is a uniform distribution, the Bayesian estimator is obtained by maximizing the likelihood function f ( x 1 , x 2 , … , x n ∣ θ ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n}\mid \theta )} . Thus

11832-490: The probabilities of testimonies, tables of mortality, judgments of tribunals, etc. which are unlikely candidates for classical probability. In this view, Poisson's contribution was his sharp criticism of the alternative "inverse" (subjective, Bayesian) probability interpretation. Any criticism by Gauss or Laplace was muted and implicit. (However, note that their later derivations of least squares did not use inverse probability.) Major contributors to "classical" statistics in

11948-1217: The probability densities that form the basis of a particular likelihood function. These conditions were first established by Chanda. In particular, for almost all x {\textstyle x} , and for all θ ∈ Θ , {\textstyle \,\theta \in \Theta \,,} ∂ log ⁡ f ∂ θ r , ∂ 2 log ⁡ f ∂ θ r ∂ θ s , ∂ 3 log ⁡ f ∂ θ r ∂ θ s ∂ θ t {\displaystyle {\frac {\partial \log f}{\partial \theta _{r}}}\,,\quad {\frac {\partial ^{2}\log f}{\partial \theta _{r}\partial \theta _{s}}}\,,\quad {\frac {\partial ^{3}\log f}{\partial \theta _{r}\,\partial \theta _{s}\,\partial \theta _{t}}}\,} exist for all r , s , t = 1 , 2 , … , k {\textstyle \,r,s,t=1,2,\ldots ,k\,} in order to ensure

12064-477: The probability density at x j {\textstyle x_{j}} amounts to maximizing the likelihood of the specific observation x j {\textstyle x_{j}} . In measure-theoretic probability theory , the density function is defined as the Radon–Nikodym derivative of the probability distribution relative to a common dominating measure. The likelihood function

12180-624: The probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation x {\textstyle x} , but not with the parameter θ {\textstyle \theta } . In the context of parameter estimation, the likelihood function is usually assumed to obey certain conditions, known as regularity conditions. These conditions are assumed in various proofs involving likelihood functions, and need to be verified in each particular application. For maximum likelihood estimation,

12296-557: The probability of two heads on two flips is P ( HH ∣ p H = 0.3 ) = 0.3 2 = 0.09. {\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.3)=0.3^{2}=0.09.} Hence L ( p H = 0.3 ∣ HH ) = 0.09. {\displaystyle {\mathcal {L}}(p_{\text{H}}=0.3\mid {\text{HH}})=0.09.} More generally, for each value of p H {\textstyle p_{\text{H}}} , we can calculate

12412-456: The probability that θ {\textstyle \theta } is the truth, given the observed sample X = x {\textstyle X=x} . Such an interpretation is a common error, with potentially disastrous consequences (see prosecutor's fallacy ). Let X {\textstyle X} be a discrete random variable with probability mass function p {\textstyle p} depending on

12528-495: The probability that the sun will rise tomorrow . Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines "out of infinitely many worlds one is selected at random ..." Little imagination is required to construct such a model, but it appears both uninteresting and meaningless. The frequentist view may have been foreshadowed by Aristotle , in Rhetoric , when he wrote:

12644-450: The probable is that which for the most part happens — Aristotle Rhetoric Poisson (1837) clearly distinguished between objective and subjective probabilities. Soon thereafter a flurry of nearly simultaneous publications by Mill , Ellis (1843) and Ellis (1854), Cournot (1843), and Fries introduced the frequentist view. Venn (1866, 1876, 1888) provided a thorough exposition two decades later. These were further supported by

12760-598: The product of univariate density functions : The goal of maximum likelihood estimation is to find the values of the model parameters that maximize the likelihood function over the parameter space, that is Intuitively, this selects the parameter values that make the observed data most probable. The specific value   θ ^ = θ ^ n ( y ) ∈ Θ   {\displaystyle ~{\hat {\theta }}={\hat {\theta }}_{n}(\mathbf {y} )\in \Theta ~} that maximizes

12876-401: The publications of Boole and Bertrand . By the end of the 19th century the frequentist interpretation was well established and perhaps dominant in the sciences. The following generation established the tools of classical inferential statistics (significance testing, hypothesis testing and confidence intervals) all based on frequentist probability. Alternatively, Bernoulli understood

12992-525: The range 0.0 to 1.0. For a perfectly fair coin , p H = 0.5 {\textstyle p_{\text{H}}=0.5} . Imagine flipping a fair coin twice, and observing two heads in two tosses ("HH"). Assuming that each successive coin flip is i.i.d. , then the probability of observing HH is P ( HH ∣ p H = 0.5 ) = 0.5 2 = 0.25. {\displaystyle P({\text{HH}}\mid p_{\text{H}}=0.5)=0.5^{2}=0.25.} Equivalently,

13108-455: The same value of θ {\displaystyle \theta } as does the maximum of L n   . {\displaystyle \,{\mathcal {L}}_{n}~.} If ℓ ( θ ; y ) {\displaystyle \ell (\theta \,;\mathbf {y} )} is differentiable in Θ , {\displaystyle \,\Theta \,,} sufficient conditions for

13224-399: The so-called posterior probability of the parameter given the observed data, which is calculated via Bayes' rule . The likelihood function, parameterized by a (possibly multivariate) parameter θ {\textstyle \theta } , is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given

13340-506: The so-called profile likelihood : The MLE is also equivariant with respect to certain transformations of the data. If y = g ( x ) {\displaystyle y=g(x)} where g {\displaystyle g} is one to one and does not depend on the parameters to be estimated, then the density functions satisfy and hence the likelihood functions for X {\displaystyle X} and Y {\displaystyle Y} differ only by

13456-415: The terms of the order ⁠ 1 /   n   ⁠  . It is possible to continue this process, that is to derive the third-order bias-correction term, and so on. However, the maximum likelihood estimator is not third-order efficient. A maximum likelihood estimator coincides with the most probable Bayesian estimator given a uniform prior distribution on the parameters . Indeed,

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