  Let $\beta\in(1,2)$ be a Pisot number and let $H_\beta$ denote Garsia's
entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963
showed that $H_\beta<1$ for any Pisot $\beta$. For the Pisot numbers which
satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been
evaluated with high precision by Alexander and Zagier and later improved by
Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other
numerical values for $H_\beta$ are known.
  In the present paper we show that $H_\beta>0.81$ for all Pisot $\beta$, and
improve this lower bound for certain ranges of $\beta$. Our method is
computational in nature.
