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The conditional Jensen inequality states that for an integrable $X$ and convex $\varphi$, whenever the terms are integrable,For differentiable $\varphi$, the supporting-line inequality $\varphi (x)\ge \varphi (m)+{\varphi }^{\prime }(m)(x-m)$ with $m=\mathbb{E}[X\mid \mathcal{G}]$ gives the result after conditional expectation. Approximation by supporting affine functions proves the general convex case.

Let $M=\mathbb{E}[Y\mid \mathcal{G}]$ and choose the stated strictly convex differentiable $\varphi$ with $|\varphi (x)|\le |x|$. Conditional Jensen inequality gives $\varphi (M)\le \mathbb{E}[\varphi (Y)\mid \mathcal{G}]$. Since $M\stackrel{d}{=}Y$, both sides have the same expectation, so equality holds almost surely. Strictness of the supporting-line inequality on $\{Y\ne M\}$ then forces $Y=M$ almost surely.

The limiting conditional error of one-nearest-neighbour classification is $1-{\sum }_k{p}_k(x{)}^2$. Put $M={\max }_k{p}_k(x)$ and $r=1-M$. By the Cauchy-Schwarz inequality, the other $K-1$ probabilities satisfyConsequentlyNow $\mathbb{E}r=R_{\mathrm{Bayes}}$, and Jensen inequality gives $\mathbb{E}{r}^2\ge (\mathbb{E}r{)}^2$. Taking expectations proves

Disintegrate $Q$ successively asFor each history $y_{<i}$, choose an optimal coupling of $P_i$ and $Q_i(\cdot \mid y_{<i})$. Sampling these couplings recursively produces a joint law of $(X,Y)$ with $Y\sim Q$. Its conditional $X_i$-marginal is always $P_i$ and is independent of the past, so $X\sim P_1\otimes \cdots \otimes P_N=P$. It is therefore a coupling $\pi \in \Pi (P,Q)$.

Write $c_i(y_{<i})$ for the conditional expected cost $w(X_i,Y_i)$ in the chosen coordinate coupling. The assumed one-coordinate transport-entropy inequality givesBy Jensen inequality and the chain rule for relative entropy,Taking the infimum over all couplings proves the tensorization of a transport-entropy inequality.

The pushforward measure of $\mu$ under $g$ is $g_{\ast }\mu$, defined for $B\in \mathcal{B}$ byThe Lebesgue decomposition theorem says that if $\nu$ and $\mu$ are sigma-finite measures on the same measurable space, then uniquely

Let $f:[0,\infty )\to \mathbb{R}\cup +\infty$ be convex with $f(1)=0$. If $\lambda$ dominates the probability distributions $P,Q$, with densities $p,q$, their f-divergence isusing the lower-semicontinuous perspective convention where $q=0$. This definition is independent of the dominating measure.

The data processing inequality for f-divergences statesTo prove it, take $\lambda =P+Q$ and let $\mathcal{G}=g^{-1}(\mathcal{B})$. If $p=dP/d\lambda$ and $q=dQ/d\lambda$, then the pullbacks of the densities of $g_{\ast }P$ and $g_{\ast }Q$ with respect to $g_{\ast }\lambda$ are respectively ${\mathbb{E}}_{\lambda }[p\mid \mathcal{G}]$ and ${\mathbb{E}}_{\lambda }[q\mid \mathcal{G}]$. The perspective $F(a,b)=bf(a/b)$ of a convex function is jointly convex. Conditional Jensen inequality therefore givesIntegration proves the claim.

The Squared Hellinger distance isIf $P,Q$ have densities $p,q$ with respect to a sigma-finite measure $\mu$, this becomes

Fix any probability distribution $Q$ and set $p_j=P_j(A_j)$, $q_j=Q(A_j)$, and $t_j=H^2(P_j,Q)$. Applying the data processing inequality to the indicator function of $A_j$ gives the Bernoulli Hellinger boundThe hinted inequality impliesThe function $t\mapsto \sqrt{t(1-t/4)}$ is concave on $[0,2]$. Since the $A_j$ form a set partition, ${\sum }_j{q}_j=1$, and Jensen inequality givesTaking the infimum over $Q$ proves the stated inequality.

Writing $P,Q$ for the laws of $X,Y$, respectively, and using Jensen inequality for the concave natural logarithm,This is the lower-bound half of the Gibbs variational principle for relative entropy.

Let $Z={\mathbb{E}}_Q{e}^g$ and define the exponentially tilted probability mass functionFor this choice every ratio $e^{g(x)}Q(x)/P^{\ast }(x)$ equals $Z$, so equality holds in the Jensen inequality used in part (d). Henceand $P^{\ast }$ is the maximizer. This is the finite-alphabet Gibbs variational principle for relative entropy.

For an i.i.d. sample and an integrable function class $\mathcal{F}$,Introduce an independent ghost sample, replace each expectation by its ghost-sample average using Jensen inequality, and then multiply each paired difference by an independent Rademacher sign. Splitting the resulting supremum into its two sample contributions gives the factor two.

In independent bond percolation on a graph, let $c_n$ be the number of $n$-step self-avoiding walks from a fixed graph vertex and ${\kappa }_n$ the number that are open. Then $\mathbb{E}{\kappa }_n=c_n{p}^n$. The Jensen inequality for the concave function $t^{1/n}$ gives $\mathbb{E}({\kappa }_n^{1/n})\le p{c}_n^{1/n}$. No independence between different self-avoiding walk indicators is needed. When $c_n^{1/n}\to \mu$, the limit superior is at most $p\mu$.

If $U$ is concave and $\mathbb{E}Z=0$, thenis concave in $s$. By Jensen inequality, $G(s)\le G(0)$; concavity then makes $G$ nonincreasing on $[0,\infty )$.

For $u\ge 0$ and a positivity-preserving operator, this is the negative log-likelihood for independent Poisson observations with intensity $1+Tu$, up to constants depending only on the data and the unit background. For bounded $g\ge 0$, its scalar derivatives are $1-g/(1+s)$ and $g/(1+s{)}^2$ on $s=Tu\ge 0$. It is convex, and strictly convex in the predicted intensity when $g>0$ almost everywhere. On a unit-area domain, the Jensen inequality gives $D_g(u)\ge \Vert Tu{\Vert }_1-\Vert g{\Vert }_{\infty }\log (1+\Vert Tu{\Vert }_1)$. The shift makes $\log (1+Tu)$ integrable whenever $Tu\in L^1$; no integrability of $\log u$ is needed.

If each probability measure ${\mu }_i$ has Poincaré constant $C_i$, then the product measure ${⨂}_i{\mu }_i$ has Poincaré constant at most ${\max }_i{C}_i$. This follows by iterating the law of total variance and applying Jensen inequality to derivatives of conditional expectations.

If each coordinate law satisfies the same convex transport-entropy inequality, then the product law satisfies the sum-cost version. Sequentially couple conditional coordinates, apply Jensen inequality, and use the chain rule for relative entropy.

For a convergent continuous local martingale and $r>1$, factor $\mathcal{E}(X)$ into $\mathcal{E}(rX{)}^{1/r^2}$ and an ordinary exponential. Apply the Holder inequality and then the concave Jensen inequality to the fractional power $2/(r+1)$. The displayed lower bound is informative when the terminal exponential moment is finite.

Cloning a vertex replaces it by several nonadjacent vertices with identical neighbourhoods. In homomorphism density, conditioning on the images of all other vertices reduces cloning to the inequality $\mathbb{E}{p}^t\ge (\mathbb{E}p{)}^t$ from the Jensen inequality. Iterated cloning shows that positive clique homomorphism density forces any fixed complete multipartite blow-up.