I'm reading the paper, T. van Erven and P. Harremoës, Rényi Divergence and Kullback-Leibler Divergence, arXiv 1206.2459 on the Renyi divergence, and I'm trying to make sense of "Example 1". I think that I'm just getting messed up on the terminology, so I'm not able to derive that the identity is true.
Am I missing some assuptions/definitions/notational conventions so that this doesn't make sense?
Is there a different way of describing/deriving their "Example 1"?
Example 1: Let $Q$ be a probability distribution and $A$ a set with positive probability.
Let $P$ be the conditional distribution of $Q$ given $A$.
Then $$ D_\alpha( P \Vert Q ) = - \ln Q(A). $$
Some features of their notation (from earlier in the paper): $$ P=(p_1, p_2, \dots, p_n) \\ Q=(q_1, q_2, \dots, q_n) $$ refer to (discrete) probability distributions.
The Renyi divergence itself is expressed as: $$ D_\alpha ( P \Vert Q ) = \frac{1}{1-\alpha} \ln \sum_{i=1}^{n} p_i^{\alpha} q_i^{1-\alpha} $$
I don't understand the notion of "the conditional distribution of $Q$ given $A$", to me, this reads as "the probability distribution of a probability distribution [function]".
I'm also unclear on the interpretation the result: the result should be the number, but what's written (looks like it) is the log of a set (note the definition of $A$).