Bayesian solution of an inverse problem for indirect measurement $M = AU + $
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 $U$.
{{journal.titleEn}} 
DownLoad: