  Once we have a measurement, how do we extract this information from the
signal? This chapter focusses on the inference of the interesting astrophysics
and cosmology once we obtain a detection of the 21-cm signal.
  Essentially, inference of the astrophysics can be broken down into three
parts:
  1. Characterisation of the observed data: The observed 21-cm signal varies
spatially as well as along the line-of-sight (frequency or redshift dimension)
to provide a full three dimensional movie of the intergalactic medium in the
early Universe. However, we cannot perform a full pixel-by-pixel comparison
between theoretical models and the observed signal. Instead, we require a
variety of statistical methods to average the observational data in order to be
able to better characterise and compare the behaviour of the faint signal.
  2. An efficient method to model the 21-cm signal: In order to interpret the
observations and understand the astrophysical processes responsible, we must be
able to produce physically motivated models capable of replicating the signal.
Further, these must be as computationally efficient as possible in order to be
able to realistically investigate the 21-cm signal.
  3. A robust probabilistic framework to extract the physics: The observed
21-cm signal is dependent on numerous physical processes, which within our
models or simulations are described by many unknown parameters. Further, these
contain approximations in order to deal with the requisite dynamic range. We
must be able to characterise our ignorance in a meaningful way in order to be
truly able to infer the astrophysical processes of the epoch of reionisation
and cosmic dawn.
  In this chapter we will focus on each separately, discussing the current
state-of-the-art in inferring astrophysical and cosmological information from
the 21cm signal.
