  A nonlinear generalisation of Schrodinger's equation is obtained using
information-theoretic arguments. The nonlinearities are controlled by an
intrinsic length scale and involve derivatives to all orders thus making the
equation mildly nonlocal. The nonlinear equation is homogeneous, separable,
conserves probability, but is not invariant under spacetime symmetries.
Spacetime symmetries are recovered when a dimensionless parameter is tuned to
vanish, whereby linearity is simultaneously established and the length scale
becomes hidden. It is thus suggested that if, in the search for a more basic
foundation for Nature's Laws, an inference principle is given precedence over
symmetry requirements, then the symmetries of spacetime and the linearity of
quantum theory might both be emergent properties that are intrinsically linked.
Supporting arguments are provided for this point of view and some testable
phenomenological consequences highlighted. The generalised Klien-Gordon and
Dirac equations are also studied, leading to the suggestion that nonlinear
quantum dynamics with intrinsically broken spacetime symmetries might be
relevant to understanding the problem of neutrino mass(lessness) and
oscillations: Among other observations, this approach hints at the existence of
a hidden discrete family symmetry in the Standard Model of particle physics.
