  We develop a new theoretical framework to generate Besan\c{c}on Galaxy Model
fast approximate simulations (BGM FASt) to address fundamental questions of the
Galactic structure and evolution performing multi-parameter inference. As a
first application of our strategy we simultaneously infer the IMF, the star
formation history and the stellar mass density in the Solar Neighbourhood. The
BGM FASt strategy is based on a reweighing scheme, that uses a specific
pre-sampled simulation, and on the assumption that the distribution function of
the generated stars in the Galaxy can be described by an analytical expression.
To validate BGM FASt we execute a set of tests. Finally, we use BGM FASt with
an approximate Bayesian computation algorithm to obtain the posterior PDF of
the inferred parameters, by comparing synthetic versus Tycho-2 colour-magnitude
diagrams. Results: The validation shows a very good agreement between BGM FASt
and the standard BGM, with BGM FASt being $\approx 10^4$ times faster. By
analysing Tycho-2 data we obtain a thin disc star formation history decreasing
in time and a present rate of $1.2 \pm 0.2 M_\odot/yr$. The resulting total
stellar mass density in the Solar Neighbourhood is $0.051_{-0.005}^{+0.002}
M_\odot/pc^3$ and the local dark matter density is $0.012 \pm 0.001
M_\odot/pc^3$. For the composite IMF we obtain a slope of
$\alpha_2={2.1}_{-0.3}^{+0.1}$ in the mass range between $0.5 M_\odot$ and
$1.53M_\odot$. The results of the slope at the high mass range are trustable up
to $4M_\odot$ and highly depend on the choice of the extinction map (obtaining
$\alpha_3={2.9}_{-0.2}^{+0.2}$ and $\alpha_3={3.7}_{-0.2}^{+0.2}$ respectively,
for two different extinction maps). Systematic uncertainties are not included.
Conclusions: The good performance of BGM FASt demonstrates that it is a very
valuable tool to perform multi-parameter inference using Gaia data releases.
