  Bayesian approach to inverse problems is studied in the case where the
forward map is a linear hypoelliptic pseudodifferential operator and
measurement error is additive white Gaussian noise. The measurement model for
an unknown Gaussian random variable $U(x,\omega)$ is \begin{eqnarray*}
M(y,\omega) = A(U(x,\omega) )+ \delta\hspace{.2mm}\mathcal{E}(y,\omega),
\end{eqnarray*} where $A$ is a finitely many times smoothing linear
hypoelliptic operator and $\delta>0$ is the noise magnitude. The covariance
operator $C_U$ of $U$ is $2r$ times smoothing, self-adjoint, injective and
elliptic pseudodifferential operator.
  If $\mathcal{E}$ was taking values in $L^2$ then in Gaussian case solving the
conditional mean (and maximum a posteriori) estimate is linked to solving the
minimisation problem \begin{eqnarray*} T_\delta(M) = \text{argmin}_{u\in H^r}
  \big\{\|A u-m\|_{L^2}^2+ \delta^2\|C_U^{-1/2}u\|_{L^2}^2 \big\}.
\end{eqnarray*} However, Gaussian white noise does not take values in $L^2$ but
in $H^{-s}$ where $s>0$ is big enough. A modification of the above approach to
solve the inverse problem is presented, covering the case of white Gaussian
measurement noise. Furthermore, the convergence of conditional mean estimate to
the correct solution as $\delta\rightarrow 0$ is proven in appropriate function
spaces using microlocal analysis. Also the contraction of the confidence
regions is studied.
