# Probability of history occurring at time $t$

I am reading LS book "Recursive Macroeconomics Theory".

In this book, as well as in some online lecture notes on the Internet. They often denote, for example, $$\pi_t(s^t)$$ for probability that history $$s^t$$ happens at time $$t$$.

But I wonder, if we are mentioning $$s^t$$, it implies time is $$t$$. So, why don't we just make it simple by writing $$\pi(s^t)$$ instead?

Thank in advance for any help!

You are right, it is the same thing. Actually, later on the book the density over the history $$s^t = [s_t, s_{t-1}, \cdots, s_0]$$ is written as
$$\pi(s^t) = \pi(s_t|s_{t-1}) \cdots \pi (s_1|s_0)\pi(s_0) \tag{2.3.1}$$
where $$\pi(s_0)$$ denotes the probability of the initial state (or $$\pi_0(s_0)$$ if you wish), and $$\pi(s|s')$$ is a transition probability.
Note that Eq. (2.3.1) encapsulates the the probability of obtaining a given history in a Markov process, since $${\rm Prob}(s_t|s_{t-1}, s_{t-2}\cdots,s_0) = {\rm Prob}(s_t|s_{t-1}) = \pi(s_t|s_{t-1})$$. This just to show that $$\pi_t(s^t)\color{blue}{\equiv}\pi(s^t)$$