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With the simplified form of the binomial for n equal to k: ∫(0 -> x)(p^n)dp = ∫(x -> 1)(p^n)dp [p^(n+1)/(n+1)](0 -> x) = [p^(n+1)/(n+1)](x -> 1) x^

by yarg 2y ago

With the simplified form of the binomial for n equal to k: ∫(0 -> x)(p^n)dp = ∫(x -> 1)(p^n)dp [p^(n+1)/(n+1)](0 -> x) = [p^(n+1)/(n+1)](x -> 1) x^(n+1) - 0^(n+1) = 1^(n+1) - x^(n+1) 2 * x^(n+1) = 1 x^(n+1) = 1/2 x = 1/(2^(1/(n+1))) Was I wrong? Have I been wrong for decades now?

hervature 2y ago
Math is more than symbolic manipulation. You need to explain where the equations come from. The best I can decipher is that you are calculating the median of a Binomial distribution given k=0. Your "error" comes from the fact that you are positing that all biases are equally likely. Thus, you should not be surprised that you are getting a non-zero probability because you yourself are saying they are non-zero.
- yarg 2y ago
  I'll write some code to test it tomorrow, but why the hell would it be zero? There are other probabilities that can lead to zero successes - every probability except for one. And yes, they diminish rapidly as the number of trials goes up (much in the same way as it does in my equation).
  - hervature 2y ago
    You have me confused a little because your original post seems to suggest you wanted it to equal 0: > I was uncomfortable with the use of biases to assign non-zero probabilities to events that fail to occur after some number of trials. Anyway, it entirely depends on your world view. When you say "There are other probabilities that can lead to zero successes" then that sounds like a Bayesian framework and that you have to pick your prior on the world. A natural choice is uniform (also Beta(1,1)) and update your prior as you collect data. You would then use the mean/median/mode of your posterior as your estimate for the bias. In your case, it appears you are operating in a Bayesian world but forcing your prior to be constantly uniform despite observing data. The frequentist perspective is that the probability is the relative frequency of observations. In this example, a frequentist would say that the probability of heads is 1 and tails is 0.
    - yarg 2y ago
      > In your case, it appears you are operating in a Bayesian world but forcing your prior to be constantly uniform despite observing data. Got you now, thanks.

by yarg 2y ago

hervature 2y ago

Math is more than symbolic manipulation. You need to explain where the equations come from. The best I can decipher is that you are calculating the median of a Binomial distribution given k=0. Your "error" comes from the fact that you are positing that all biases are equally likely. Thus, you should not be surprised that you are getting a non-zero probability because you yourself are saying they are non-zero.

yarg 2y ago
I'll write some code to test it tomorrow, but why the hell would it be zero? There are other probabilities that can lead to zero successes - every probability except for one. And yes, they diminish rapidly as the number of trials goes up (much in the same way as it does in my equation).
- hervature 2y ago
  You have me confused a little because your original post seems to suggest you wanted it to equal 0: > I was uncomfortable with the use of biases to assign non-zero probabilities to events that fail to occur after some number of trials. Anyway, it entirely depends on your world view. When you say "There are other probabilities that can lead to zero successes" then that sounds like a Bayesian framework and that you have to pick your prior on the world. A natural choice is uniform (also Beta(1,1)) and update your prior as you collect data. You would then use the mean/median/mode of your posterior as your estimate for the bias. In your case, it appears you are operating in a Bayesian world but forcing your prior to be constantly uniform despite observing data. The frequentist perspective is that the probability is the relative frequency of observations. In this example, a frequentist would say that the probability of heads is 1 and tails is 0.
  - yarg 2y ago
    > In your case, it appears you are operating in a Bayesian world but forcing your prior to be constantly uniform despite observing data. Got you now, thanks.