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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?

Bjarke Kingo
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  • It heavily depends on what else you know about the function. If you know that it's smooth, you can extrapolate a little from the points you have, but only a little. There's no general way to extrapolate a function only known at a handful of points without some model behind it – Yuriy S Oct 24 '18 at 11:49
  • Sorry, it's too vague. We need to know more about the function. – lulu Oct 24 '18 at 11:49
  • With no assumptions, you cannot estimate any probability.

    Before you can start talking about the probability of "something", you need to know what the probability space is.

     – 5xum Oct 24 '18 at 11:49
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    If there's context for this question - more information about where it comes from, whether it's measuring something - you can [edit] the question to tell us more. Then perhaps we can help. Otherwise the answer is "no" as other comments and answers say. – Ethan Bolker Oct 24 '18 at 12:01
  • This is a completely different question (after the edit). You should try to reflect that in the title, as well as the tags – Yuriy S Oct 24 '18 at 12:07

4 Answers4

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With no assumptions, you cannot estimate any probability.

Before you can start talking about the probability of "something", you need to know what the probability space is, and possibly, at least a little bit about how the probabilities of events are distributed.

For example, if you know that your function is a cubic polynomial, then you can answer with certainty whether it will increase in "future". No other input is needed here.

If you know that the function is, for example, a fifth order polynomial, then you need some assumptions on how the coefficients of the function are distributed, otherwise you cannot answer the question.

If you only know the function is... well... a function, then your question has way too little information to answer.

5xum
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If there is a causal relation between the successive function values, you can hope for reliable extrapolation. For instance by linear prediction.

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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.

Ethan Bolker
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The answer is no. You can only work with what you know about your function. What you can do is use regression analysis and find a curve which fits your data and hope that the curve will represent your function for other values as well. That is what they do in statistics and numerical analysis but you need a large set of data to be satisfied with your results.

Mohammad Riazi-Kermani
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