# Questions tagged [log-concave]

Probability distributions on the Euclidean space having a concave log-density and generalizations of these.

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### Log-concavity of a discrete random distribution

I'm working with a discrete distribution on the set of non-negative integers. The step-wise cumulative distribution function at any non-negative integer $i$ is  F_i = \frac{2}{\log w}[\log(1+w^{i+1})...
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### For log-concave densities, are joint and marginal modes consistent?

Suppose I have a probability density function $\pi(x_1, \ldots, x_n)$, which is the density of a vector-valued random variable $X$ in $\mathbb{R}^n$. Assume that $\pi$ is strongly log-concave, i.e., ...
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### Mixture of Gaussian is not log-concave

I've encountered the statement: For $p\in(0,1),$ the location mixture of standard univariate normal densities $f(x)=p\phi(x)+(1-p)\phi(x-\mu)$ is log-concave if and only if $\Vert\mu\Vert \leq 2.$ ...
• 213
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### What is global concavity of the (log-)likelihood worth in Bayesian estimation?

In maximum likelihood estimation there is a big emphasis on finding the global maximum, which is why likelihood functions that are provably globally (log-)concave are desirable (despite often being ...
• 1,406
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### Bound on the expectation of a function of random variable having a strictly log-concave probability density

let $\theta \in \mathbb{R}^d$ be a random variable having a strictly log-concave probability density function, i.e $$p(\theta) = e^{-\phi(\theta)}$$ where $\phi(\theta)$ is ...
• 21
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### Joint negative log-likelihood of the Binomial model $\mathrm{Binomial}(n, p)$ with unknown $n$ and $p$

I'm trying to make sense of the following plot portraying the joint negative log-likelihood for the Binomial model $\mathrm{Binomial}(n,p)$ with unknown $n$ and $p$ and $y=5$. Note that the white part ...
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Let $X$ and $Y$ be two random variables (or vectors) with continuous and "smooth enough" joint distribution. Assume that the two conditional distributions $X|Y=y$ and $Y|X=x$ are log-concave for all \$...