  In this paper, we provide an algorithm to estimate from below the dimension
of self-similar measures with overlaps. As an application, we show that for any
$ \beta\in(1,2) $, the dimension of the Bernoulli convolution $ \mu_\beta $
satisfies
  \[ \dim (\mu_\beta) \geq 0.9804085,\]
  which improves a previous uniform lower bound $0.82$ obtained by Hare and
Sidorov \cite{HareSidorov2018}. This new uniform lower bound is very close to
the known numerical approximation $ 0.98040931953\pm 10^{-11}$ for $\dim
\mu_{\beta_3}$, where $ \beta_{3} \approx 1.839286755214161$ is the largest
root of the polynomial $ x^{3}-x^{2}-x-1$.
  Moreover, the infimum $\inf_{\beta\in (1,2)}\dim (\mu_\beta) $ is attained at
a parameter $\beta_*$ in a small interval
  \[ (\beta_{3} -10^{-8}, \beta_{3} + 10^{-8}).\]
  When $\beta$ is a Pisot number, we express $\dim(\mu_\beta)$ in terms of the
measure-theoretic entropy of the equilibrium measure for certain matrix
pressure function, and present an algorithm to estimate $\dim (\mu_\beta)$ from
above as well.
