Your yeti revenue distribution would predict a maximum of 76 at day 0, but we see 77. We also don’t see anything below 73. I suggest 72 + 1d5 at day 0.
The full formula would be then 72 + 1d5 - {age}d6, where these are independent d6 rolls, and not a single roll multiplied by the age (I think this is what you meant, but clarifying since I wasn’t 100% sure on the terminology used). (The other interpretation would lead to too high variation at high ages, I think).
The formulas for Snow Serpent and Winter Wolf look consistent with the data to me.
Yeah I debated that, and some other similar modifications, but in the end my priors on what the natural laws of these worlds look like overrode what looks like it might be a better fit. We shall see...
:facepalm: that’s what I get for switching at the last minute. You’re absolutely right, of course.
I originally had 78 - {age+1}d6 but I figured there was no way that was actually the original formula. I don’t particularly like 71 + 1d6 - {age}d6 either. The means make “-{age}d6” look really good, but I don’t see any “nice” way to start it off. 72+1d5, like you said, looks great distribution-wise but 72 < {20,25} and 1d5 < {all the d6s}. 71+1d6 takes a hit on never rolling a 1 but gets a boost from 1d6 vs 1d5.
OK I calculated the likelihoods of getting the full yeti revenue data set for those distributions and got the following results:
For 78-{age+1}d6 (which is equivalent to 71+1d6-{age}d6):
likelihood ~ 10^-192
For 72 + 1d5 - {age}d6:
likelihood ~10^-189
So the 1d5 version is literally 1000 times better fit to the data, and I doubt that the prior for 1d6 over 1d5 is that strong. Besides, the base value of 72 is a round number in a way, might increase the 1d5 prior a bit. I’d definitely bet on the 1d5 version over the 1d6.
That’s pretty convincing. Let’s try all the maybe-reasonable-ish versions:
71+1d6-{age}d6 has log[10] likelihood of −192.02 72+1d6-{age}d6 has log[10] likelihood of −190.57 72+1d5-{age}d6 has log[10] likelihood of −188.91 71+1d6-{age}d5 has log[10] likelihood of −196.42 72+1d6-{age}d5 has log[10] likelihood of −206.39 72+1d5-{age}d5 has log[10] likelihood of −200.41
Definitively -{age}d6, sure. Now here’s a much tighter question: what about 72+1d6-{age}d6? It takes a x45 hit on the likelihood compared to 72+1d5-{age}d6, which is pretty big, but I think I’m willing to say the chances of using a lone 1d5 right there are a bigger penalty. So my revised guess is 72+1d6-{age}d6.
Your yeti revenue distribution would predict a maximum of 76 at day 0, but we see 77. We also don’t see anything below 73. I suggest 72 + 1d5 at day 0.
The full formula would be then 72 + 1d5 - {age}d6, where these are independent d6 rolls, and not a single roll multiplied by the age (I think this is what you meant, but clarifying since I wasn’t 100% sure on the terminology used). (The other interpretation would lead to too high variation at high ages, I think).
The formulas for Snow Serpent and Winter Wolf look consistent with the data to me.
Yeah I debated that, and some other similar modifications, but in the end my priors on what the natural laws of these worlds look like overrode what looks like it might be a better fit. We shall see...
And yes, I was using standard https://en.wikipedia.org/wiki/Dice_notation
OK, I see such an argument for the die used, but the base value proposed can’t be correct with that die.
:facepalm: that’s what I get for switching at the last minute. You’re absolutely right, of course.
I originally had 78 - {age+1}d6 but I figured there was no way that was actually the original formula. I don’t particularly like 71 + 1d6 - {age}d6 either. The means make “-{age}d6” look really good, but I don’t see any “nice” way to start it off. 72+1d5, like you said, looks great distribution-wise but 72 < {20,25} and 1d5 < {all the d6s}. 71+1d6 takes a hit on never rolling a 1 but gets a boost from 1d6 vs 1d5.
OK I calculated the likelihoods of getting the full yeti revenue data set for those distributions and got the following results:
For 78-{age+1}d6 (which is equivalent to 71+1d6-{age}d6):
likelihood ~ 10^-192
For 72 + 1d5 - {age}d6:
likelihood ~10^-189
So the 1d5 version is literally 1000 times better fit to the data, and I doubt that the prior for 1d6 over 1d5 is that strong. Besides, the base value of 72 is a round number in a way, might increase the 1d5 prior a bit. I’d definitely bet on the 1d5 version over the 1d6.
That’s pretty convincing. Let’s try all the maybe-reasonable-ish versions:
71+1d6-{age}d6 has log[10] likelihood of −192.02
72+1d6-{age}d6 has log[10] likelihood of −190.57
72+1d5-{age}d6 has log[10] likelihood of −188.91
71+1d6-{age}d5 has log[10] likelihood of −196.42
72+1d6-{age}d5 has log[10] likelihood of −206.39
72+1d5-{age}d5 has log[10] likelihood of −200.41
Definitively -{age}d6, sure. Now here’s a much tighter question: what about
72+1d6-{age}d6?
It takes a x45 hit on the likelihood compared to
72+1d5-{age}d6, which is pretty big, but I think I’m willing to say the chances of using a lone 1d5 right there are a bigger penalty. So my revised guess is 72+1d6-{age}d6.
Nice analysis! But I am not so confident on how strong the prior should be and am somewhat torn on the conclusion.