Can i determine if a function will increase in the future? Is it possible to determine whether a function will have increased in the future relative to starting points, given a sample of the first $m$ points?
For example, given the 4 first values of a function $(f(1), f(2), f(3), f(4) )= (2, 1, 4, 5 )$ can I with some probability determine whether the $f(n)$ will be larger than all $f(1)$ to $f(4)$.
I apologize if this is too vague. Any answers appreciated
EDIT:
Thanks for commenting. If i let my function be something like a ratio of how many white and black marbles i have in my collection, and i keep adding marbles (non-fair with no known probability), can i then pursue something?

Based on your edit, I will venture a guess at the underlying question. I think you may be filling an urn one marble at a time where each time there is a fixed unknown probability $p$ that the new ball is black. You observe the first few marbles, and want to know something about the long term behavior.
At that point your best estimate of $p$ is the last ratio of black to total that you have seen. With that value of $p$ you can use standard techniques to calculate probabilities (not certainties) for questions about the future. As you see more and more data your estimate of $p$ and your estimate of its uncertainty will improve.
