= Exponential tilting
{wiki}
= Exponentially tilted
{synonym}
<exponential tilting> reweights a <probability law> in favour of larger or smaller values. Let $K(\theta)=\log\mathbb E_Pe^{\theta X}$ be finite at the chosen parameter. Define the tilted law by its <Radon-Nikodym derivative>
$$
\frac{dP_\theta}{dP}(x)=e^{\theta x-K(\theta)}.
$$
Its total mass is one by construction. Positive $\theta$ favours larger $x$, while negative $\theta$ favours smaller $x$; it changes <probabilities> without creating values outside the original <support of a probability distribution>[support].
For $\theta$ in the interior of the finite moment-generating-function domain, differentiation under the <integral> gives
$$
\mathbb E_\theta X=K'(\theta),\qquad
\operatorname{Var}_\theta(X)=K''(\theta).
$$
The interior-domain condition supplies the <integrable> exponential bounds needed for these <derivatives>. Endpoint tilts can exist without a finite tilted <mean>, so existence of a normalized tilt alone is insufficient for the <mean> formula.
For a <Bernoulli distribution> with success <probability> $p\in(0,1)$, the tilted success <probability> is $p_\theta=pe^\theta/(1-p+pe^\theta)$: tilting adds $\theta$ to its <log odds>. A <Normal distribution> $N(\mu,\sigma^2)$ tilts to $N(\mu+\sigma^2\theta,\sigma^2)$ by completing the square. These calculations explain why <exponential tilting> is useful for making a rare <Sample mean> typical while retaining an explicit <Likelihood ratio>.
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