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Comment by ribrars | Hacker News Reader

> How do you arrive at that 5% base chance? I actually don’t know an analytic way of arriving there, but you can get there experimentally pretty easily by just writing a loop that runs through this a million times and then tells you how often it succeeded or failed. Then you build a lookup table that contains the percentages.

I'm no stats expert by any means, but couldn't you calculate the percentages of each event occuring (and the amount to increment) using bayesian modeling?

Simply taking the product of a string of actions and calculating those seems like an acceptable solution, however I'm not sure.

The author seems to use a calculation approach, which if I understand correctly is a valid method and is sampling the distribution. But then again, not sure.

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An easy way to do the odds analytically is to calculate the average expected number of trials needed for a success, and then note that the overall average success rate is the inverse of that.

E.g. in the author's example, where the rate starts at 5% and increases by 5% with each failure, the expected number of trials needed for a success would be:

    sum = 1 * 0.05 +                // hit the first trial
          2 * 0.95 * 0.10 +         // missed first, hit second
          3 * (0.95*0.9) * 0.15 +   // missed first two, hit third
          // ..and so on

Summing up that loop gives 5.29 for the expected number of trials, the inverse of which is 18.9%, matching the author's observed result.

The problem is the inverse: starting with 18.9%, how do you get the 5+5%

The author doesn't examine that question. They comment that one can run a large number of trials to estimate the 18.9 given the 5+5, and my post is how to get that answer analytically for any sequence of probabilities.

For the reverse case, there are arbitrarily many such sequences that would average out to any given value. In practice someone using this technique for gamedev wouldn't necessarily only want X+X%.

Imagine a binary tree. The root node represents the probability of 5%. If you take the left child to mean a positive result, the probability stays at 5%. If you take the right child to mean a negative result the probability rises to 10%.

Every left child produces a tree from that point which is exactly the same at the root tree.

Every right child produces a probability of +5% until you get to 100% after which the right node is forced back to 5% — meaning it will match the root from that point.

I feel like once you’ve defined the tree to depth 20, you could average all the probabilities and it should equal 19%, because everything from that depth further is just a uniform repetition of the probability tree, but it’s probably slightly more complicated than that.

> I'm no stats expert by any means, but couldn't you calculate the percentages of each event occuring (and the amount to increment) using bayesian modeling?

No. Probability theory is about working forward from a given model to a set of possible observations. And Bayesian statistics is about working backwards from given observations to which model in a set of models produced them (which is why an old name for Bayesian statistics is 'inverse statistics').

In this case, there is no given set of observations; there's not some game out there which has produced a dataset and you're trying to work backwards to figure out what model that game is using. Instead, you have a bunch of models and you're trying to work forwards to figure out which of them will give you the kind of outputs you want. So, that's probability theory.

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