  We examine the effect of a prior that favours low values of fine-tuning on
Bayesian multi-dimensional fits of the constrained minimal supersymmetric
standard model (CMSSM or mSUGRA) to current data. The dark matter relic
density, the anomalous magnetic moment of the muon and the branching ratio of
b->s gamma are all used to constrain the model via a Markov Chain Monte Carlo
sampler. As a result of the naturalness prior, posterior probability
distributions skew towards lighter higgs and sparticle masses, the effect being
most pronounced in the gaugino sector. Interestingly, slepton masses are an
exception and skew towards heavier masses. The lightest CP-even Higgs h^0-pole
region becomes allowed at the 2 sigma level for the latest combination of
measurements of m_t=172.7+/-2.9 GeV, provided we allow for a theoretical error
in the prediction of its mass m_{h^0}. m_{h^0} is constrained to be less than
120 GeV at the 95% C.L. Probing the branching ratio of B_s->mu^+ mu^- to the
level of 2 x 10^{-8}, as might be achieved by the Tevatron experiments, would
cover 32% of the probability density, irrespective of which of the two priors
is used.
