  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.

High Energy Astrophysical Phenomena;Solar and Stellar Astrophysics