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Logarithmically concave function

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In <a href="Convex_analysis" title="Convex analysis">convex analysis</a>, a <a href="Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> function f : Rn → R+ is logarithmically concave (or log-concave for short) if its <a href="Domain_of_a_function" title="Domain of a function">domain</a> is a <a href="Convex_set" title="Convex set">convex set</a>, and if it satisfies the inequality

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<annotation encoding="application/x-tex">{\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }}</annotation>

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</math><img src="./124e7078828b9179cbfa5ab37c1f735f1318d0b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.273ex; height:3.176ex;" alt="{\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }}" loading="lazy"></dd></dl>

for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the <a href="Logarithm" title="Logarithm">logarithm</a> of the function, log ∘ f, is <a href="Concave_function" title="Concave function">concave</a>; that is,

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<annotation encoding="application/x-tex">{\displaystyle \log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)}</annotation>

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</math><img src="./b0f46590c9ac42a32050411021390b646a273e25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.85ex; height:2.843ex;" alt="{\displaystyle \log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)}" loading="lazy"></dd></dl>

for all x,y ∈ dom f and 0 < θ < 1.

Examples of log-concave functions are the 0-1 <a href="Indicator_function" title="Indicator function">indicator functions</a> of convex sets (which requires the more flexible definition), and the <a href="Gaussian_function" title="Gaussian function">Gaussian function</a>.

Similarly, a function is <a href="Log-convex" class="mw-redirect" title="Log-convex">log-convex</a> if it satisfies the reverse inequality

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<annotation encoding="application/x-tex">{\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }}</annotation>

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</math><img src="./f65174e0bd63da908694d8ce38a49a17d3cbce5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.273ex; height:3.176ex;" alt="{\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }}" loading="lazy"></dd></dl>

for all x,y ∈ dom f and 0 < θ < 1.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<ul><li>A log-concave function is also <a href="Quasi-concave_function" class="mw-redirect" title="Quasi-concave function">quasi-concave</a>. This follows from the fact that the logarithm is monotone implying that the <a href="Level_set#Sublevel_and_superlevel_sets" title="Level set">superlevel sets</a> of this function are convex.<a href="#cite_note-:0-1">[1]</a></li>

<li>Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the <a href="Gaussian_function" title="Gaussian function">Gaussian function</a> f(x) = exp(−x2/2) which is log-concave since log f(x) = −x2/2 is a concave function of x. But f is not concave since the <a href="Second_derivative" title="Second derivative">second derivative</a> is positive for |x| > 1:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0}</annotation>

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</math><img src="./ff35561e82e5ca53ec648a7ce07e24a1d0c33b1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.147ex; height:4.676ex;" alt="{\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0}" loading="lazy"></dd></dl></dd></dl>

<ul><li>From above two points, <a href="Concave_function" title="Concave function">concavity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

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<annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation>

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</math><img src="./469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"> log-concavity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

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<annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation>

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</math><img src="./469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"> <a href="Quasiconcave_function" class="mw-redirect" title="Quasiconcave function">quasiconcavity</a>.</li>

<li>A twice differentiable, nonnegative function with a convex domain is log-concave <a href="If_and_only_if" title="If and only if">if and only if</a> for all x satisfying f(x) > 0,</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}}</annotation>

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</math><img src="./1335715979d10dfeabc218f9479a6a3c6c544522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.02ex; height:3.176ex;" alt="{\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}}" loading="lazy">,<a href="#cite_note-:0-1">[1]</a></dd></dl></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}}</annotation>

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</math><img src="./25a859c6474bb1ec33eb89fcb872133e5fdab11d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.762ex; height:3.176ex;" alt="{\displaystyle f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}}" loading="lazy"> is</dd></dl></dd></dl>

<dl><dd><a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">negative semi-definite</a>. For functions of one variable, this condition simplifies to</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle f(x)f''(x)\leq (f'(x))^{2}}</annotation>

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</math><img src="./baadd589d12d98ebf0d5f59614c07ac3c337fa16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.12ex; height:3.176ex;" alt="{\displaystyle f(x)f''(x)\leq (f'(x))^{2}}" loading="lazy"></dd></dl></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Operations_preserving_log-concavity">Operations preserving log-concavity</h2></div>

<ul><li>Products: The product of log-concave functions is also log-concave. Indeed, if f and g are log-concave functions, then log f and log g are concave by definition. Therefore</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \log \,f(x)+\log \,g(x)=\log(f(x)g(x))}</annotation>

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</math><img src="./15187575cc9dc021aa35b21659789183d53cd3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.557ex; height:2.843ex;" alt="{\displaystyle \log \,f(x)+\log \,g(x)=\log(f(x)g(x))}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>is concave, and hence also f g is log-concave.</dd></dl>

<ul><li><a href="Marginal_distribution" title="Marginal distribution">Marginals</a>: if f(x,y) : Rn+m → R is log-concave, then</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle g(x)=\int f(x,y)dy}</annotation>

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</math><img src="./c8492cadd1b18c2d4d9b9ad4996e3ed9147577f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.913ex; height:5.676ex;" alt="{\displaystyle g(x)=\int f(x,y)dy}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>is log-concave (see <a href="Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a>).</dd></dl>

<ul><li>This implies that <a href="Convolution" title="Convolution">convolution</a> preserves log-concavity, since h(x,y) = f(x-y) g(y) is log-concave if f and g are log-concave, and therefore</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy}</annotation>

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</math><img src="./90d0fe78c53b4233fe2b78a0d5883c5a907facbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.8ex; height:5.676ex;" alt="{\displaystyle (f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>is log-concave.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Log-concave_distributions">Log-concave distributions</h2></div>

Log-concave distributions are necessary for a number of algorithms, e.g. <a href="Adaptive_rejection_sampling" class="mw-redirect" title="Adaptive rejection sampling">adaptive rejection sampling</a>. Every distribution with log-concave density is a <a href="Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy probability distribution</a> with specified mean μ and <a href="Deviation_risk_measure" title="Deviation risk measure">Deviation risk measure</a> D.<a href="#cite_note-Grechuk1-2">[2]</a>

As it happens, many common <a href="Probability_distribution" title="Probability distribution">probability distributions</a> are log-concave. Some examples:<a href="#cite_note-:1-3">[3]</a>

<ul><li>the <a href="Normal_distribution" title="Normal distribution">normal distribution</a> and <a href="Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distributions</a>,</li>

<li>the <a href="Exponential_distribution" title="Exponential distribution">exponential distribution</a>,</li>

<li>the <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> over any <a href="Convex_set" title="Convex set">convex set</a>,</li>

<li>the <a href="Binomial_distribution" title="Binomial distribution">binomial distribution</a>,</li>

<li>the <a href="Logistic_distribution" title="Logistic distribution">logistic distribution</a>,</li>

<li>the <a href="Extreme_value_distribution" class="mw-redirect" title="Extreme value distribution">extreme value distribution</a>,</li>

<li>the <a href="Laplace_distribution" title="Laplace distribution">Laplace distribution</a>,</li>

<li>the <a href="Chi_distribution" title="Chi distribution">chi distribution</a>,</li>

<li>the <a href="Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">hyperbolic secant distribution</a>,</li>

<li>the <a href="Wishart_distribution" title="Wishart distribution">Wishart distribution</a>, if n ≥ p + 1,<a href="#cite_note-prekopa-4">[4]</a></li>

<li>the <a href="Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a>, if all parameters are ≥ 1,<a href="#cite_note-prekopa-4">[4]</a></li>

<li>the <a href="Gamma_distribution" title="Gamma distribution">gamma distribution</a> if the <a href="Shape_parameter" title="Shape parameter">shape parameter</a> is ≥ 1,</li>

<li>the <a href="Chi-square_distribution" class="mw-redirect" title="Chi-square distribution">chi-square distribution</a> if the number of degrees of freedom is ≥ 2,</li>

<li>the <a href="Beta_distribution" title="Beta distribution">beta distribution</a> if both shape parameters are ≥ 1, and</li>

<li>the <a href="Weibull_distribution" title="Weibull distribution">Weibull distribution</a> if the shape parameter is ≥ 1.</li></ul>

Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.

The following distributions are non-log-concave for all parameters:

<ul><li>the <a href="Student's_t-distribution" title="Student's t-distribution">Student's t-distribution</a>,</li>

<li>the <a href="Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>,</li>

<li>the <a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a>,</li>

<li>the <a href="Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a>, and</li>

<li>the <a href="F-distribution" title="F-distribution">F-distribution</a>.</li></ul>

Note that the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:

<ul><li>the <a href="Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a>,</li>

<li>the <a href="Pareto_distribution" title="Pareto distribution">Pareto distribution</a>,</li>

<li>the <a href="Weibull_distribution" title="Weibull distribution">Weibull distribution</a> when the shape parameter < 1, and</li>

<li>the <a href="Gamma_distribution" title="Gamma distribution">gamma distribution</a> when the shape parameter < 1.</li></ul>

The following are among the properties of log-concave distributions:

<ul><li>If a density is log-concave, so is its <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF).</li>

<li>If a multivariate density is log-concave, so is the <a href="Marginal_density" class="mw-redirect" title="Marginal density">marginal density</a> over any subset of variables.</li>

<li>The sum of two independent log-concave <a href="Random_variable" title="Random variable">random variables</a> is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.</li>

<li>The product of two log-concave functions is log-concave. This means that <a href="Joint_distribution" class="mw-redirect" title="Joint distribution">joint</a> densities formed by multiplying two probability densities (e.g. the <a href="Normal-gamma_distribution" title="Normal-gamma distribution">normal-gamma distribution</a>, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purpose <a href="Gibbs_sampling" title="Gibbs sampling">Gibbs sampling</a> programs such as <a href="Bayesian_inference_using_Gibbs_sampling" title="Bayesian inference using Gibbs sampling">BUGS</a> and <a href="Just_another_Gibbs_sampler" title="Just another Gibbs sampler">JAGS</a>, which are thereby able to use <a href="Adaptive_rejection_sampling" class="mw-redirect" title="Adaptive rejection sampling">adaptive rejection sampling</a> over a wide variety of <a href="Conditional_distribution" class="mw-redirect" title="Conditional distribution">conditional distributions</a> derived from the product of other distributions.</li>

<li>If a density is log-concave, so is its <a href="Survival_function" title="Survival function">survival function</a>.<a href="#cite_note-:1-3">[3]</a></li>

<li>If a density is log-concave, it has a monotone <a href="Hazard_rate" class="mw-redirect" title="Hazard rate">hazard rate</a> (MHR), and is a <a href="Regular_distribution_(economics)" title="Regular distribution (economics)">regular distribution</a> since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\log \left(1-F(x)\right)=-{\frac {f(x)}{1-F(x)}}}</annotation>

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</math><img src="./2f3f5c103dfe30a94f09533b220174e2900fbbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.058ex; height:6.509ex;" alt="{\displaystyle {\frac {d}{dx}}\log \left(1-F(x)\right)=-{\frac {f(x)}{1-F(x)}}}" loading="lazy"> which is decreasing as it is the derivative of a concave function.</dd></dl></dd></dl>

<ul><li><a href="Logarithmically_concave_sequence" title="Logarithmically concave sequence">logarithmically concave sequence</a></li>

<li><a href="Logarithmically_concave_measure" title="Logarithmically concave measure">logarithmically concave measure</a></li>

<li><a href="Logarithmically_convex_function" title="Logarithmically convex function">logarithmically convex function</a></li>

<li><a href="Convex_function" title="Convex function">convex function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFBoydVandenberghe2004" class="citation book cs1"><a href="Stephen_P._Boyd" title="Stephen P. Boyd">Boyd, Stephen</a>; Vandenberghe, Lieven (2004). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/">"Log-concave and log-convex functions"</a>. Convex Optimization. Cambridge University Press. pp. 104–108. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-83378-7</bdi>.</cite>

</li>

<li id="cite_note-Grechuk1-2"><a href="#cite_ref-Grechuk1_2-0">^</a> <cite id="CITEREFGrechukMolybohaZabarankin2009" class="citation journal cs1">Grechuk, Bogdan; Molyboha, Anton; Zabarankin, Michael (May 2009). <a rel="nofollow" class="external text" href="https://www.researchgate.net/profile/Bogdan-Grechuk/publication/220442393_Maximum_Entropy_Principle_with_General_Deviation_Measures/links/59132b61a6fdcc963e7ed4fd/Maximum-Entropy-Principle-with-General-Deviation-Measures.pdf">"Maximum Entropy Principle with General Deviation Measures"</a> (PDF). <a href="Mathematics_of_Operations_Research" title="Mathematics of Operations Research">Mathematics of Operations Research</a>. 34 (2): 445–467. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmoor.1090.0377">10.1287/moor.1090.0377</a>.</cite>

</li>

<li id="cite_note-:1-3">^ <a href="#cite_ref-:1_3-0">a</a> <a href="#cite_ref-:1_3-1">b</a> See <cite id="CITEREFBagnoliBergstrom2005" class="citation journal cs1">Bagnoli, Mark; Bergstrom, Ted (2005). <a rel="nofollow" class="external text" href="http://www.econ.ucsb.edu/~tedb/Theory/delta.pdf">"Log-Concave Probability and Its Applications"</a> (PDF). Economic Theory. 26 (2): 445–469. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00199-004-0514-4">10.1007/s00199-004-0514-4</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1046688">1046688</a>.</cite>

</li>

<li id="cite_note-prekopa-4">^ <a href="#cite_ref-prekopa_4-0">a</a> <a href="#cite_ref-prekopa_4-1">b</a> <cite id="CITEREFPrékopa1971" class="citation journal cs1"><a href="Andr%C3%A1s_Pr%C3%A9kopa" title="András Prékopa">Prékopa, András</a> (1971). <a rel="nofollow" class="external text" href="http://rutcor.rutgers.edu/~prekopa/SCIENT1.pdf">"Logarithmic concave measures with application to stochastic programming"</a> (PDF). <a href="Acta_Scientiarum_Mathematicarum" title="Acta Scientiarum Mathematicarum">Acta Scientiarum Mathematicarum</a>. 32 (3–4): 301–316.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li><cite id="CITEREFDharmadhikariJoag-Dev1988" class="citation book cs1">Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-214690-5</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0954608">0954608</a>.</cite></li></ul>

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<ul><li><cite id="CITEREFPečarićProschanTong1992" class="citation book cs1">Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-549250-2</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1162312">1162312</a>.</cite></li></ul></div><div><div class="zim-footer">

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