  A general shape identification inverse problem is studied in a Bayesian
framework. This problem requires the determination of the unknown shape of a
domain in the Euclidean space from finite-dimensional observation data with
some Gaussian random noise. Then, the stability of posterior is studied for
observation data. For each point of the space, the conditional probability that
the point is included in the unknown domain given the observation data is
considered. The stability is also studied for this probability distribution. As
a model problem for our inverse problem, a heat inverse problem is considered.
This problem requires the determination of the unknown shape of cavities in a
heat conductor from temperature data of some portion of the surface of the heat
conductor. To apply the above stability results to this model problem, one
needs the measurability and some boundedness of the forward operator. These
properties are shown.
