[This is the second piece in my series on the Parametric Process Assumption (PPA), a new model of anthropics. You should start here if you are just seeing this for the first time.
TLDR:
This series will use a visualization called "mosaic plots"
Mosaic plots represent outcomes in a probability distribution as boxes, with relative sizes representing the probabilities of different outcomes
Mosaic plots can be conditioned by throwing out all boxes that don't meet some condition
We show Mosaic plots for the SSA, SIA, and explain that they are "indexical" because they assign probability to being a particular observer
A non-indexical distribution we call "Updateless Double Halfer," all in the Sleeping Beauty Problem
A tool we will be using a lot in this series is what is known as a mosaic plot. A mosaic plot is just an illustration of a probability distribution with a bunch of boxes, where each box corresponds to some outcome and the size of each box is proportional to the probability of that outcome. The roll of a six sided die, for instance, would look like this:
Here, you can see that each outcome (1 through 6) is equally likely, as all boxes are the same size.
The main value of mosaic plots for us is that they make conditioning very easy. For instance, if the only thing you knew was that the die had come up even, then you could remove all odd boxes and get a new mosaic plot that shows only the even ones as options.
In this new chart, we have three equally likely outcomes: 2, 4, and 6. From these two charts we could conclude that, for example, P(4) = 1/6 and P(4 | even) = 1/3.
In terms of anthropics, if we were to apply the SSA to the Sleeping Beauty problem, it would look like this:
Notice how each outcome covers both a universe and an observer assignment - that is because this distribution is “indexical,” meaning that it assigns probabilities to both worlds and identities within those worlds. We can see that “Heads” and “Tails” are equally likely, but “Heads + Monday” is twice as likely as “Tails + Monday”. If we learned it was in fact Monday, we would get the following mosaic plot where Heads was twice as likely as tails:
This chart is no longer rectangular, and most of our charts won’t be. It’s the relative size of the boxes, not their arrangement, which matters.
Anyway, just to round it out, this is the mosaic for the SIA. As you can see, it begins with a 1/3 probability of Heads, but when you learn it is Monday the probability goes back to 50%:
This chart actually establishes a convention which we will use on all mosaic plots throughout this series to make them more interpretable. When our distribution is indexical, our mosaic plots will always be divided into rows where each one represents a different universe outcome (Heads or Tails in this case) and the height of the row exactly corresponds to the universe prior of that state. In the case of the SIA and PPA, everything will also be vertically aligned, and as such all columns will correspond to a specific observer identity (here, Monday is the first column and Tuesday is the second). This type of alignment won’t be possible with SSA, but we’ll still relegate observer identities to the x-axis.
Finally, because we’re going to discuss it more in a future piece, here is a mosaic plot for a “Double Halfer” method inspired by Ape in the Coat’s work and that of others.[1] I will be referring to it as “Updateless Double Halfer” (UDH) in this series, since as far as I know it doesn’t have another more specific name and I want to differentiate it from other Double Halfer methods which we will not be covering, like Bostrom’s Hybrid Model.
Instead of asking “which wake-up is currently happening?” the UDH asks “which wake-ups occur at any point in our experiment?” As such, the outcome “Tails” automatically implies both Monday and Tuesday, and our chart is really just showing us our non-indexical prior:
It is updateless because, upon learning that it is Monday, it can update only on “Monday happens”, which occurs for both Heads and Tails and does not change anything from the prior.
I say “inspired by” because while I’m confident that this mosaic plot aligns with his description of the solution to the Sleeping Beauty Problem, but it's unclear to me if he would agree with its generalization to other situations.
[This is the second piece in my series on the Parametric Process Assumption (PPA), a new model of anthropics. You should start here if you are just seeing this for the first time.
TLDR:
The next piece is "A Brief Defense of Indexical Reasoning."]
A tool we will be using a lot in this series is what is known as a mosaic plot. A mosaic plot is just an illustration of a probability distribution with a bunch of boxes, where each box corresponds to some outcome and the size of each box is proportional to the probability of that outcome. The roll of a six sided die, for instance, would look like this:
Here, you can see that each outcome (1 through 6) is equally likely, as all boxes are the same size.
The main value of mosaic plots for us is that they make conditioning very easy. For instance, if the only thing you knew was that the die had come up even, then you could remove all odd boxes and get a new mosaic plot that shows only the even ones as options.
In this new chart, we have three equally likely outcomes: 2, 4, and 6. From these two charts we could conclude that, for example, P(4) = 1/6 and P(4 | even) = 1/3.
In terms of anthropics, if we were to apply the SSA to the Sleeping Beauty problem, it would look like this:
Notice how each outcome covers both a universe and an observer assignment - that is because this distribution is “indexical,” meaning that it assigns probabilities to both worlds and identities within those worlds. We can see that “Heads” and “Tails” are equally likely, but “Heads + Monday” is twice as likely as “Tails + Monday”. If we learned it was in fact Monday, we would get the following mosaic plot where Heads was twice as likely as tails:
This chart is no longer rectangular, and most of our charts won’t be. It’s the relative size of the boxes, not their arrangement, which matters.
Anyway, just to round it out, this is the mosaic for the SIA. As you can see, it begins with a 1/3 probability of Heads, but when you learn it is Monday the probability goes back to 50%:
This chart actually establishes a convention which we will use on all mosaic plots throughout this series to make them more interpretable. When our distribution is indexical, our mosaic plots will always be divided into rows where each one represents a different universe outcome (Heads or Tails in this case) and the height of the row exactly corresponds to the universe prior of that state. In the case of the SIA and PPA, everything will also be vertically aligned, and as such all columns will correspond to a specific observer identity (here, Monday is the first column and Tuesday is the second). This type of alignment won’t be possible with SSA, but we’ll still relegate observer identities to the x-axis.
Finally, because we’re going to discuss it more in a future piece, here is a mosaic plot for a “Double Halfer” method inspired by Ape in the Coat’s work and that of others.[1] I will be referring to it as “Updateless Double Halfer” (UDH) in this series, since as far as I know it doesn’t have another more specific name and I want to differentiate it from other Double Halfer methods which we will not be covering, like Bostrom’s Hybrid Model.
Instead of asking “which wake-up is currently happening?” the UDH asks “which wake-ups occur at any point in our experiment?” As such, the outcome “Tails” automatically implies both Monday and Tuesday, and our chart is really just showing us our non-indexical prior:
It is updateless because, upon learning that it is Monday, it can update only on “Monday happens”, which occurs for both Heads and Tails and does not change anything from the prior.
The next piece is "A Brief Defense of Indexical Reasoning"
I say “inspired by” because while I’m confident that this mosaic plot aligns with his description of the solution to the Sleeping Beauty Problem, but it's unclear to me if he would agree with its generalization to other situations.