  A method is developed for fitting theoretically predicted astronomical
spectra to an observed spectrum. Using a hierarchical Bayesian principle, the
method takes both systematic and statistical measurement errors into account,
which has not been done before in the astronomical literature. The goal is to
estimate fundamental stellar parameters and their associated uncertainties. The
non-availability of a convenient deterministic relation between stellar
parameters and the observed spectrum, combined with the computational
complexities this entails, necessitate the curtailment of the continuous
Bayesian model to a reduced model based on a grid of synthetic spectra. A
criterion for model selection based on the so-called predictive squared error
loss function is proposed, together with a measure for the goodness-of-fit
between observed and synthetic spectra. The proposed method is applied to the
infrared 2.38--2.60 \mic ISO-SWS data (Infrared Space Observatory - Short
Wavelength Spectrometer) of the star $\alpha$ Bootis, yielding estimates for
the stellar parameters: effective temperature \Teff = 4230 $\pm$ 83 K, gravity
$\log$ g = 1.50 $\pm$ 0.15 dex, and metallicity [Fe/H] = $-0.30 \pm 0.21$ dex.
