= White-noise likelihood for a square-integrable shift
{title2=$L(h,m)=\exp(W_m(h)-\|h\|_2^2/2)$}
Generalized <Gaussian white noise> over a real <Hilbert space> $H=L^2$ is an <isonormal Gaussian process> $W(h)$ with covariance $\langle h,g\rangle_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued <Gaussian measure> because it is not a <trace-class operator>. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with <Radon-Nikodym derivative>
$$
\frac{dP_h}{dP_0}(m)=\exp\left(W_m(h)-\tfrac12\|h\|_H^2\right).
$$
In an <orthonormal basis>, $W_m(h)=\sum_jm_jh_j$ is a stochastic series with total <variance> $\|h\|_H^2$, not an inner product of two $H$-valued observations. The formula follows from the <Cameron-Martin theorem for a Gaussian measure> in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On $\mathbb R^2$, generalized noise can be realized on <tempered distributions>; an unweighted global negative <Sobolev space> is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.
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