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**ε**is considered, where $U$ is a function on a domain of $\R^d$. Here $A$ is a smoothing linear operator and

**ε**is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k$

**ε**, where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k$

**ε**. Bayes formula gives then the posterior distribution

$\pi_{kn}(u_n\|\m_{kn})$~ Π _{n} $(u_n)\exp(-\frac{1}{2}$||$\m_{kn} - P_kA u_n$||$\_2^2)$

in $\R^d$,
and the mean $\u_{kn}$:$=\int
u_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing
prior distributions Π _{n } for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case.
Such choice of prior distributions is * discretization-invariant* in the sense that Π _{n } represent the same * a priori* information for all $n$ and that the mean $\u_{kn}$ converges to a limit estimate as $k,n$→$\infty$.
Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_11$ prior is related to penalizing the $\l^1$ norm of the wavelet coefficients of $U$.

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