  In view of the recent Fermi observations of GRB prompt emission spectra, we
develop a theory of photosphere emission of a hybrid relativistic outflow with
a hot fireball component (defined by dimensionless entropy $\eta$) and a cold
Poynting-flux component (defined by magnetization $\sigma_0$ at the central
engine). We consider the scenarios both without and with sub-photospheric
magnetic dissipations. Based on a simplified toy model of jet dynamics, we
develop two approaches: a "bottom-up" approach to predict the temperature (for
a non-dissipative photosphere) and luminosity of the photosphere emission and
its relative brightness for a given pair of $(\eta,\sigma_0)$; and a "top-down"
approach to diagnose central engine parameters ($\eta$ and $\sigma_0$) based on
the observed quasi-thermal photosphere emission properties. We show that a
variety of observed GRB prompt emission spectra with different degrees of
photosphere thermal emission can be reproduced by varying $\eta$ and $\sigma_0$
within the non-dissipative photosphere scenario. In order to reproduce the
observed spectra, the outflows of most GRBs need to have a significant
$\sigma$, both at the central engine, and at the photosphere. The $\sigma$
value at $10^{15}$ cm from the central engine (a possible non-thermal emission
site) is usually also greater than unity, so that internal-collision-induced
magnetic reconnection and turbulence (ICMART) may be the mechanism to power the
non-thermal emission. We apply our top-down approach to GRB 110721A, and find
that the temporal evolution behavior of its blackbody component can be well
interpreted with a time-varying $(\eta,\sigma_0)$ at the central engine,
instead of invoking a varying engine base size $r_0$ as proposed by previous
authors.
