I have an (optimization) algorithm that, on every iteration, outputs an integer number. Output is non-increasing; plotting the output wrt. cycle index shows that the relation from output to cycles looks roughly like the right hand of an hyperbola (or inverse exponential) that tends towards some optimal value $b$. I.e., if $x$ it the cycle index and $y$ the output number, a "fitting curve" could be either:
$y = a·x^{-k} + b$
or
$y = e^{a-kx} + b$
for some constants $a$, $k$ and $b$. Note that $x$ cannot be negative and for all that matters, $y$ can be considered undefined for $x=0$.
What I want is to be able to predict in the early iterations, what the value of $b$ will be, together with some probability. For example, at the $x$th iteration, I want to be able to ask "what is the probability that $b$ is less than 100?" I expect the answer to be very imprecise during the first iterations (i.e., will give very high probabilities for any query), and tends towards extreme answers (either 0 or 1) when $x$ tends towards infinity.
At first sight (under the hypothesis that the inverse exponential model is correct), it looks like computing an exponential regression. However, exponential regression models that I found usually consider that there is no such thing as a $b$ constant, and I have never found the way to compute a probability from a regression.
I have taken probability, statistics and numerical analysis courses years ago, but I have never heard of such a tool. Is this standard matter?
Here is an excerpt of my data:
cycle output
1 552
2 532
3 513
4 492
5 480
6 451
7 420
8 405
9 383
10 365
…
32 176
33 176
34 175
35 171
36 171
…
536 171
537 167
538 154
539 154
…
2135 154
2136 151
2137 146
…
3000 146