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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 [https://en.wikipedia.org/wiki/Complex_logarithm complex logarithm] is the multi-valued [https://en.wikipedia.org/wiki/Inverse_function inverse] of the complex exponential function. Similarly, the [https://en.wikipedia.org/wiki/Discrete_logarithm discrete logarithm] is the multi-valued inverse of the exponential function in finite groups; it has uses in [https://en.wikipedia.org/wiki/Public-key_cryptography public-key cryptography].<br/> <br/> full text link : [https://en.wikipedia.org/wiki/Logarithm https://en.wikipedia.org/wiki/Logarithm]<br/>
=== Likely hoodLikelihood === The '''likelihood function''' (often simply called the '''likelihood''') is the [https://en.wikipedia.org/wiki/Joint_probability_distribution joint] [https://en.wikipedia.org/wiki/Probability_mass_function probability mass] (or [https://en.wikipedia.org/wiki/Probability_density_function probability density]) of [https://en.wikipedia.org/wiki/Sample_(statistics) observed data] viewed as a function of the [https://en.wikipedia.org/wiki/Statistical_parameter parameters] of a [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> Intuitively, the likelihood function ๐ฟ(๐โฃ๐ฅ)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/24a053912e70a2d35f7037375a39f9f7c3ea72d4> is the probability of observing data ๐ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4> assuming ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af> is the actual parameter. In [https://en.wikipedia.org/wiki/Maximum_likelihood_estimation maximum likelihood estimation], the [https://en.wikipedia.org/wiki/Arg_max arg max] (over the parameter ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>) of the likelihood function serves as a [https://en.wikipedia.org/wiki/Point_estimation point estimate] for ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>, while the [https://en.wikipedia.org/wiki/Fisher_information Fisher information] (often approximated by the likelihood's [https://en.wikipedia.org/wiki/Hessian_matrix Hessian matrix]) indicates the estimate's [https://en.wikipedia.org/wiki/Precision_(statistics) precision]. In contrast, in [https://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics], parameter estimates are derived from the ''converse'' of the likelihood, the so-called [https://en.wikipedia.org/wiki/Posterior_probability posterior probability], which is calculated via [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/> The likelihood function, parameterized by a (possibly multivariate) parameter ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af>, is usually defined differently for [https://en.wikipedia.org/wiki/Continuous_or_discrete_variable discrete and continuous] [https://en.wikipedia.org/wiki/Probability_distribution probability distributions] (a more general definition is discussed below). Given a probability density or mass function ๐ฅโฆ๐(๐ฅโฃ๐),<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/442aa0f5b4796ef4a698a7e60aeb5006c8f020f2> where ๐ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4> is a realization of the random variable ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab>, the likelihood function is ๐โฆ๐(๐ฅโฃ๐),<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/e161b494cb41c7ddbb8d496ece959b776baba128><br/> often written<br/> ๐ฟ(๐โฃ๐ฅ).<br/> <img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/487868b15b5aaccd5bf67e86c197d68f37fadc8f> In other words, when ๐(๐ฅโฃ๐)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/0f01a2e70b1a8595be545c42562f00820bbff06d> is viewed as a function of ๐ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4> with ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af> fixed, it is a probability density function, and when viewed as a function of ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af> with ๐ฅ<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4> fixed, it is a likelihood function. In the [https://en.wikipedia.org/wiki/Frequentist_probability frequentist paradigm], the notation ๐(๐ฅโฃ๐)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/0f01a2e70b1a8595be545c42562f00820bbff06d> is often avoided and instead ๐(๐ฅ;๐)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/79480c3540803bdda2613d69277692e1061ad7d5> or ๐(๐ฅ,๐)<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3b8aafdf0be69fcd09cdb756b9c5aa2fd8c777> are used to indicate that ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af> is regarded as a fixed unknown quantity rather than as a [https://en.wikipedia.org/wiki/Random_variable random variable] being conditioned on. The likelihood function does ''not'' specify the probability that ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af> is the truth, given the observed sample ๐=๐ฅ<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 [https://en.wikipedia.org/wiki/Prosecutor's_fallacy prosecutor's fallacy]).<br/> <br/> full text link : [https://en.wikipedia.org/wiki/Likelihood_function https://en.wikipedia.org/wiki/Likelihood_function]
