Open main menu

Biolecture.org β

Changes

no edit summary
The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the&nbsp;[https://en.wikipedia.org/wiki/Complex_logarithm complex logarithm]&nbsp;is the multi-valued&nbsp;[https://en.wikipedia.org/wiki/Inverse_function inverse]&nbsp;of the complex exponential function. Similarly, the&nbsp;[https://en.wikipedia.org/wiki/Discrete_logarithm discrete logarithm]&nbsp;is the multi-valued inverse of the exponential function in finite groups; it has uses in&nbsp;[https://en.wikipedia.org/wiki/Public-key_cryptography public-key cryptography].<br/> <br/> full text link&nbsp;:&nbsp;[https://en.wikipedia.org/wiki/Logarithm https://en.wikipedia.org/wiki/Logarithm]<br/> &nbsp;
 === Likely hoodLikelihood === The&nbsp;'''likelihood function'''&nbsp;(often simply called the&nbsp;'''likelihood''') is the&nbsp;[https://en.wikipedia.org/wiki/Joint_probability_distribution joint]&nbsp;[https://en.wikipedia.org/wiki/Probability_mass_function probability mass]&nbsp;(or&nbsp;[https://en.wikipedia.org/wiki/Probability_density_function probability density]) of&nbsp;[https://en.wikipedia.org/wiki/Sample_(statistics) observed data]&nbsp;viewed as a function of the&nbsp;[https://en.wikipedia.org/wiki/Statistical_parameter parameters]&nbsp;of a&nbsp;[https://en.wikipedia.org/wiki/Statistical_model statistical model].<sup id="cite_ref-1">[https://en.wikipedia.org/wiki/Likelihood_function#cite_note-1 [1]]</sup><sup id="cite_ref-2">[https://en.wikipedia.org/wiki/Likelihood_function#cite_note-2 [2]]</sup><sup id="cite_ref-3">[https://en.wikipedia.org/wiki/Likelihood_function#cite_note-3 [3]]</sup>&nbsp;Intuitively, the likelihood function&nbsp;๐ฟ(๐œƒโˆฃ๐‘ฅ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/24a053912e70a2d35f7037375a39f9f7c3ea72d4>&nbsp;is the probability of observing data&nbsp;๐‘ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4>&nbsp;assuming&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>&nbsp;is the actual parameter. In&nbsp;[https://en.wikipedia.org/wiki/Maximum_likelihood_estimation maximum likelihood estimation], the&nbsp;[https://en.wikipedia.org/wiki/Arg_max arg max]&nbsp;(over the parameter&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>) of the likelihood function serves as a&nbsp;[https://en.wikipedia.org/wiki/Point_estimation point estimate]&nbsp;for&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>, while the&nbsp;[https://en.wikipedia.org/wiki/Fisher_information Fisher information]&nbsp;(often approximated by the likelihood's&nbsp;[https://en.wikipedia.org/wiki/Hessian_matrix Hessian matrix]) indicates the estimate's&nbsp;[https://en.wikipedia.org/wiki/Precision_(statistics) precision]. In contrast, in&nbsp;[https://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics], parameter estimates are derived from the&nbsp;''converse''&nbsp;of the likelihood, the so-called&nbsp;[https://en.wikipedia.org/wiki/Posterior_probability posterior probability], which is calculated via&nbsp;[https://en.wikipedia.org/wiki/Bayes'_theorem Bayes' rule].<sup id="cite_ref-4">[https://en.wikipedia.org/wiki/Likelihood_function#cite_note-4 [4]]</sup><br/> &nbsp; The likelihood function, parameterized by a (possibly multivariate) parameter&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>, is usually defined differently for&nbsp;[https://en.wikipedia.org/wiki/Continuous_or_discrete_variable discrete and continuous]&nbsp;[https://en.wikipedia.org/wiki/Probability_distribution probability distributions]&nbsp;(a more general definition is discussed below). Given a probability density or mass function &nbsp; ๐‘ฅโ†ฆ๐‘“(๐‘ฅโˆฃ๐œƒ),<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/442aa0f5b4796ef4a698a7e60aeb5006c8f020f2> &nbsp; where&nbsp;๐‘ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4>&nbsp;is a realization of the random variable&nbsp;๐‘‹<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab>, the likelihood function is ๐œƒโ†ฆ๐‘“(๐‘ฅโˆฃ๐œƒ),<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/e161b494cb41c7ddbb8d496ece959b776baba128><br/> often written<br/> ๐ฟ(๐œƒโˆฃ๐‘ฅ).<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/487868b15b5aaccd5bf67e86c197d68f37fadc8f> &nbsp; In other words, when&nbsp;๐‘“(๐‘ฅโˆฃ๐œƒ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/0f01a2e70b1a8595be545c42562f00820bbff06d>&nbsp;is viewed as a function of&nbsp;๐‘ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4>&nbsp;with&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>&nbsp;fixed, it is a probability density function, and when viewed as a function of&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>&nbsp;with&nbsp;๐‘ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4>&nbsp;fixed, it is a likelihood function. In the&nbsp;[https://en.wikipedia.org/wiki/Frequentist_probability frequentist paradigm], the notation&nbsp;๐‘“(๐‘ฅโˆฃ๐œƒ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/0f01a2e70b1a8595be545c42562f00820bbff06d>&nbsp;is often avoided and instead&nbsp;๐‘“(๐‘ฅ;๐œƒ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/79480c3540803bdda2613d69277692e1061ad7d5>&nbsp;or&nbsp;๐‘“(๐‘ฅ,๐œƒ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3b8aafdf0be69fcd09cdb756b9c5aa2fd8c777>&nbsp;are used to indicate that&nbsp;๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>&nbsp;is regarded as a fixed unknown quantity rather than as a&nbsp;[https://en.wikipedia.org/wiki/Random_variable random variable]&nbsp;being conditioned on. The likelihood function does&nbsp;''not''&nbsp;specify the probability that&nbsp; ๐œƒ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>&nbsp;is the truth, given the observed sample&nbsp;๐‘‹=๐‘ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/0661396d873679039ffe8e908a39f02402d4912d>. Such an interpretation is a common error, with potentially disastrous consequences (see&nbsp;[https://en.wikipedia.org/wiki/Prosecutor's_fallacy prosecutor's fallacy]).<br/> <br/> full text link&nbsp;:&nbsp;[https://en.wikipedia.org/wiki/Likelihood_function https://en.wikipedia.org/wiki/Likelihood_function]
&nbsp;
82
edits