  In this note we present an algorithm to obtain a uniform lower bound on
Hausdorff dimension of the stationary measure of an affine iterated function
scheme with similarities, the best known example of which is Bernoulli
convolution. The Bernoulli convolution measure $\mu_\lambda$ is the probability
measure corresponding to the law of the random variable $\xi =
\sum_{k=0}^\infty \xi_k\lambda^k$, where $\xi_k$ are i.i.d. random variables
assuming values $-1$ and $1$ with equal probability and $\frac12 < \lambda <
1$. In particular, for Bernoulli convolutions we give a uniform lower bound
$\dim_H(\mu_\lambda) \geq 0.96399$ for all $\frac12<\lambda<1$.
