  We consider the off-policy evaluation problem in Markov decision processes
with function approximation. We propose a generalization of the recently
introduced \emph{emphatic temporal differences} (ETD) algorithm
\citep{SuttonMW15}, which encompasses the original ETD($\lambda$), as well as
several other off-policy evaluation algorithms as special cases. We call this
framework \ETD, where our introduced parameter $\beta$ controls the decay rate
of an importance-sampling term. We study conditions under which the projected
fixed-point equation underlying \ETD\ involves a contraction operator, allowing
us to present the first asymptotic error bounds (bias) for \ETD. Our results
show that the original ETD algorithm always involves a contraction operator,
and its bias is bounded. Moreover, by controlling $\beta$, our proposed
generalization allows trading-off bias for variance reduction, thereby
achieving a lower total error.
