  The radius $R_{1.4}$ of neutron stars (NSs) with a mass of 1.4 M$_{\odot}$
has been extracted consistently in many recent studies in the literature. Using
representative $R_{1.4}$ data, we infer high-density nuclear symmetry energy
$E_{\rm{sym}}(\rho)$ and the associated nucleon specific energy $E_0(\rho)$ in
symmetric nuclear matter (SNM) within a Bayesian statistical approach using an
explicitly isospin-dependent parametric Equation of State (EOS) for nucleonic
matter. We found that: (1) The available astrophysical data can already improve
significantly our current knowledge about the EOS in the density range of
$\rho_0-2.5\rho_0$. In particular, the symmetry energy at twice the saturation
density $\rho_0$ of nuclear matter is determined to be
$E_{\mathrm{sym}}(2\rho_0)$ =39.2$_{-8.2}^{+12.1}$ MeV at 68\% confidence
level. (2) A precise measurement of the $R_{1.4}$ alone with a 4\% 1$\sigma$
statistical error but no systematic error will not improve much the constraints
on the EOS of dense neutron-rich nucleonic matter compared to what we extracted
from using the available radius data. (3) The $R_{1.4}$ radius data and other
general conditions, such as the observed NS maximum mass and causality
condition introduce strong correlations for the high-order EOS parameters.
Consequently, the high-density behavior of $E_{\rm{sym}}(\rho)$ inferred
depends strongly on how the high-density SNM EOS $E_0(\rho)$ is parameterized,
and vice versa. (4) The value of the observed maximum NS mass and whether it is
used as a sharp cut-off for the minimum maximum mass or through a Gaussian
distribution affect significantly the lower boundaries of both the $E_0(\rho)$
and $E_{\rm{sym}}(\rho)$ only at densities higher than about $2.5\rho_0$.
