Definition: Total loss of a history (with geometric time discount)
Let and let denote the time-horizon, which may be infinite. The total loss of a given history with time discount is defined by
Earlier, the momentary loss L was defined more generally, as a function of type (A \times O)^* → [0,1]. Here, it appears as if L is only a function of the current observation, as opposed to being able to take the entire history into account.
Alternatively I may have misunderstood. The alternative interpretation that comes to mind is that the total loss is only allowed to be constructed by time discounting a loss function of observations only.
where is the prefix of of length . The formula in the post is written for the special case where the momentary loss only depends on the last observation, but it should been stated in full generality.
Earlier, the momentary loss L was defined more generally, as a function of type (A \times O)^* → [0,1]. Here, it appears as if L is only a function of the current observation, as opposed to being able to take the entire history into account.
Alternatively I may have misunderstood. The alternative interpretation that comes to mind is that the total loss is only allowed to be constructed by time discounting a loss function of observations only.
You’re right, it’s supposed to be
where