Say we have an ordered list of items
[a, b, c, ... x, y, z, ...]
I am looking for a family of distributions with support on the list above governed by some parameter alpha so that:
- For alpha=0, it assigns probability 1 to the first item, a above, and 0 to the rest. That is, if we sample from this list, with replacement, we always get
- As alpha increases, we assign higher and higher probabilities to the rest of the list, respecting the ordering of the list, following ~exponential decay.
- When alpha=1, we assign equal probability to all items in the list, so sampling from the list is akin to ignoring its ordering.
This is is very similar to the geometric distribution, but there are some notable differences:
- The geometric distribution distribution is defined over all natural numbers. In my case above, the list has fixed size.
- The geometric distribution isn't defined for alpha=0.