  The smooth spin-down of young pulsars is perturbed by two non-deterministic
phenomenon, glitches and timing noise. Although the timing noise provides
insights into nuclear and plasma physics at extreme densities, it acts as a
barrier to high-precision pulsar timing experiments. An improved methodology
based on Bayesian inference is developed to simultaneously model the stochastic
and deterministic parameters for a sample of 85 high-$\dot{E}$ radio pulsars
observed for $\sim$ 10 years with the 64-m Parkes radio telescope. Timing noise
is known to be a red process and we develop a parametrization based on the
red-noise amplitude ($A_{\rm red}$) and spectral index ($\beta$). We measure
the median $A_{\rm red}$ to be $-10.4^{+1.8}_{-1.7}$ yr$^{3/2}$ and $\beta$ to
be $-5.2^{+3.0}_{-3.8}$ and show that the strength of timing noise scales
proportionally to $\nu^{1}|\dot{\nu}|^{-0.6\pm0.1}$, where $\nu$ is the spin
frequency of the pulsar and $\dot{\nu}$ its spin-down rate. Finally, we measure
significant braking indices for 19 pulsars, proper motions for two pulsars and
discuss the presence of periodic modulation in the arrival times of five
pulsars.
