Recent progress in machine learning methods and the emerging
availability
of programmable interfaces for scanning probe microscopes (SPMs) have
propelled automated and autonomous microscopies to the forefront of
attention of the scientific community. However, enabling automated
microscopy requires the development of task-specific machine learning
methods, understanding the interplay between physics discovery and
machine learning, and fully defined discovery workflows. This, in
turn, requires balancing the physical intuition and prior knowledge
of the domain scientist with rewards that define experimental goals
and machine learning algorithms that can translate these to specific
experimental protocols. Here, we discuss the basic principles of Bayesian
active learning and illustrate its applications for SPM. We progress
from the Gaussian process as a simple data-driven method and Bayesian
inference for physical models as an extension of physics-based functional
fits to more complex deep kernel learning methods, structured Gaussian
processes, and hypothesis learning. These frameworks allow for the
use of prior data, the discovery of specific functionalities as encoded
in spectral data, and exploration of physical laws manifesting during
the experiment. The discussed framework can be universally applied
to all techniques combining imaging and spectroscopy, SPM methods,
nanoindentation, electron microscopy and spectroscopy, and chemical
imaging methods and can be particularly impactful for destructive
or irreversible measurements.