  Time-distance helioseismology is a set of powerful tools to study features
below the Sun's surface. Inverse methods are needed to interpret time-distance
measurements, with many examples in the literature. However, techniques that
utilize a more statistical approach to inferences, and broadly used in the
astronomical community, are less-commonly found in helioseismology. This
article aims to introduce a potentially powerful inversion scheme based on
Bayesian probability theory and Monte Carlo sampling that is suitable for local
helioseismology. We describe the probabilistic method and how it is
conceptually different from standard inversions used in local helioseismology.
Several example calculations are carried out to compare and contrast the setup
of the problems and the results that are obtained. The examples focus on two
important phenomena studied with helioseismology: meridional circulation and
supergranulation. Numerical models are used to compute synthetic observations,
providing the added benefit of knowing the solution against which the results
can be tested. For demonstration purposes, the problems are formulated in two
and three dimensions, using both ray- and Born-theoretical approaches. The
results seem to indicate that the probabilistic inversions not only find a
better solution with much more realistic estimation of the uncertainties, but
they also provide a broader view of the range of solutions possible for any
given model, making the interpretation of the inversion more quantitative in
nature. Unlike the progress being made in fundamental measurement schemes in
local helioseismology that image the far side of the Sun, or have detected
signatures of global Rossby waves, among many others, inversions of those
measurements have had significantly less success. Such statistical methods may
help overcome some of these barriers to move the field forward.
