Constraining the dark energy equation of state using Bayes theorem and
  the Kullback-Leibler divergence
  Data-driven model-independent reconstructions of the dark energy equation of
state $w(z)$ are presented using Planck 2015 era CMB, BAO, SNIa and
Lyman-$\alpha$ data. These reconstructions identify the $w(z)$ behaviour
supported by the data and show a bifurcation of the equation of state posterior
in the range $1.5{<}z{<}3$. Although the concordance $\Lambda$CDM model is
consistent with the data at all redshifts in one of the bifurcated spaces, in
the other a supernegative equation of state (also known as `phantom dark
energy') is identified within the $1.5 \sigma$ confidence intervals of the
posterior distribution. To identify the power of different datasets in
constraining the dark energy equation of state, we use a novel formulation of
the Kullback--Leibler divergence. This formalism quantifies the information the
data add when moving from priors to posteriors for each possible dataset
combination. The SNIa and BAO datasets are shown to provide much more
constraining power in comparison to the Lyman-$\alpha$ datasets. Further, SNIa
and BAO constrain most strongly around redshift range $0.1-0.5$, whilst the
Lyman-$\alpha$ data constrains weakly over a broader range. We do not attribute
the supernegative favouring to any particular dataset, and note that the
$\Lambda$CDM model was favoured at more than $2$ log-units in Bayes factors
over all the models tested despite the weakly preferred $w(z)$ structure in the
data.
