This is the tenth post in my series on Anthropics. The previous one is Beauty and the Bets.

Introduction

In my previous posts I've been talking about the actual object-level disagreement between halfers and thirders - which of the answers formally is correct and which is not. I've shown that there is one correct model for the Sleeping Beauty problem, that describes it instead of something else, successfully passes the statistical test, has sound mathematical properties and deals with every betting scheme.

But before we can conclude that the issue is fully resolved, there is still a notable semantic disagreement left, as well as several interesting questions. If the thirder answer isn't the correct "probability", then what is it? What are the properties of this entity? And why are people so eager to confuse it with the actual probability?

In this post I'm going to answer these questions and dissolve the last, purely semantic part of the disagreement.

A Limitation of Probability Theory

Probability theory has an curious limitation. It is unable to talk about events which happen multiple times during the same iteration of an experiment. Instead, when the conditions of the experiment state that something happens multiple times, we are supposed to define "happening multiple times" as an elementary outcome in itself.

This is, actually, a very useful property, that allows probability theory to work as intended, so that the measure of an event properly corresponds to the credence that rational agents are suppose to have in that event. But it may lead to conflicts with our intuitions about the matter. 

Let's consider an example:

A coin is tossed. On Heads you are given a blue ball, on Tails you are given a blue ball and then you are given a red ball.

Here we say that there are two well-defined mutually exclusive outcomes. Exactly one of them happens at every iteration of the experiment.

$\Omega =\{Heads\&Blue,Tails\&Both\}=\{Heads\&Blue\&!Red,Tails\&Blue\&Red\}$ 

We can go into a bit more details and define separate sample spaces for getting a blue ball and getting a red ball:

$\left\{Blue\right\}$

$P\left(Blue\right)=1$

$\{Red,!Red\}$

$P\left(Red\right)=P(!Red)=1/2$

We can coherently talk about the probability to get only one ball. Such event happens when the red ball isn't given

$P\left(OnlyOne\right)=P\left(Heads\&Blue\right)=P(!Red)=1/2$

And the probability to get any ball at all. Such an event happens on every iteration of the probability experiment, as you at least get a blue ball.

$P\left(Any\right)=P\left(Heads\&Blue\right)+P\left(Tails\&Both\right)=P\left(BlueorRed\right)=1$

And if we were asked what is the probability that the coin landed Heads, conditional on the fact that we received any ball, the answer is simple:

$P\left(Heads|BlueorRed\right)=P\left(Heads\right)=1/2$

But for some people it may feel that something is off here. Doesn't the event "Getting a ball" happen twice on Tails? According to probability theory, it doesn't. Only one outcome from the sample space can happen per iteration of a probability experiment, and there is no way the same event can happen twice.

But it may feel that there has to be something that happens once when the coin is Heads and twice when the coin is Tails. Even though it's impossible to formally define as an event in the probability space, it may still feel that we should be able to talk about it! 

Defining Weighted Probability Space

Let's define a new entity indexical sample space, which has to possess all the properties of a sample space, except that it doesn't require its elements to be mutually exclusive. 

$\Omega b^{\prime }=\{Red,Blue\}$

Red/Blue means the same as previously - getting a red/blue ball during the experiment. Such outcomes couldn't define a regular sample space for our problem, because on Tails both of them happen. But we specifically defined indexical sample space to be irrelevant to this concern.

And now let's enrich our indexical sample space by the sample space of the coin toss. 

$\Omega c=\{Heads,Tails\}$

if we simply take Cartesian product of the two we get:

${\Omega }^{\prime }=\{Heads\&Blue,Heads\&Red,Tails\&Blue,Tails\&Red\}$

Here we have a bit of an issue with $Heads\&Red$ - this outcome doesn't really happen. But that's fine: we have two options, either we can just assume that corresponding elementary event has zero measure and thus we can remove this outcome from our enriched indexical sample space beforehand, or we can initially keep it and then update on the fact that it doesn't happen later. Both of these methods eventually lead to the same values of our measure function. Here, for simplicity I'll just remove it, and so we get:

${\Omega }^{\prime }=\{Heads\&Blue,Tails\&Blue,Tails\&Red\}$

Now we can define $F^{\prime }$ - indexical event space as some sigma-algebra over the indexical sample space ${\Omega }^{\prime }$ and $P^{\prime }$ - weighted probability - a measure function the domain of which is $F^{\prime }$. It's similar to regular probability function, with the only difference that instead of

$P\left(\Omega \right)=1$

We now have

$P^{\prime }\left({\Omega }^{\prime }\right)=1$

And therefore we get

$({\Omega }^{\prime },F^{\prime },P^{\prime })$ - weighted probability space.

We can look at it as a generalization of probability space. While in every iteration of a probability experiment there is only one outcome from the sample space that is realized, here we can have multiple outcomes from the indexical sample space, that are realized during the same iteration of the experiment. In our example, on Tails both Tails&Blue and Tails&Red are realized. 

Properties of Weighted Probability Function

The weighted probability function gives us the probability of an event happening, weighted by the number of outcomes from the indexical sample space that can happen during one iteration of the experiment. And so, the weighted probability that the coin is Heads conditionally on the fact that any ball was received is:

$P^{\prime }\left(Heads|BlueorRed\right)=P^{\prime }\left(Heads\right)=1/3$

This, mind you, in no way contradicts the fact that $P\left(Heads\right)=1/2$. $P$ and $P^{\prime }$ are, generally speaking, two different functions and, therefore, can produce different results when receiving the same arguments.

Conservation of Expected Evidence

Neither should we be troubled by the fact that unconditional weighted probability of a fair coin is not 1/2. For a regular probability function that would be a paradoxical situation, because the unconditional probability of a coin being Heads depends fully on the fairness of the coin. But weighted probability also depends on the number of events that can happen during one iteration of experiment.

Instead of following Conservation of Expected Evidence, $P^{\prime }$ follows a new principle which we can call Conservation of Expected Evidence and Weight. According to which, a weighted probability estimate can be changed either due to receiving new evidence, or when the number of the outcomes from the indexical sample space that can be realized per one iteration of experiment changes. 

A consequence from this principle is that, if both the number of outcomes changes and new evidence is received in a compensatory way, weighted probability stays the same.

Relation to Probability

Switching from probability to weighted probability is easy. We simply need to renormalize the measure function so that $P^{\prime }\left({\Omega }^{\prime }\right)=1$.

In our example

$P\left(Blue\right)=1$

$P\left(Red\right)=1/2$

and 

$P^{\prime }\left(Blue\right)+P^{\prime }\left(Red\right)=1$

so

$P^{\prime }\left(Blue\right)=P\left(Blue\right)\left(P^{\prime }\right(Blue)+P^{\prime }(Red\left)\right)/\left(P\right(Blue)+P(Red\left)\right)=2/3$

$P^{\prime }\left(Red\right)=1-P^{\prime }\left(Blue\right)=1/3$

Essentially, weighted probability function treats some of the non-mutually exclusive events the way probability function treats mutually exclusive events. So if we, for some reason, confuse weighted probability with probability, we will be talking about a different problem, where events $Blue$ and $Red$ indeed are mutually exclusive:

You are given blue ball 2/3 of time and red ball 1/3 of time. Half the time you get the Blue ball the coin is put Tails, in other situations it's put Heads.

Domain of Function

As you might have noticed, I've deliberately selected an example without any memory loss. For the sake of simplicity, but also to explicitly show that amnesia is irrelevant to the question whether we can use the framework of weighted probabilities or not.

All we need is the ability to formally define a weighted probability space and, as it has less strict limitations than a regular probability space, we can at least always do it when a probability space is defined.

In the trivial cases, a weighted probability space is the exact same thing as a regular probability space:

$({\Omega }^{\prime },F^{\prime },P^{\prime })=(\Omega ,F,P)$

In the more interesting cases, when we have something to weight the probabilities by, as in the example above, the situation is different, but whether we have a trivial case or not doesn't depend on the participant of the experiment going through amnesia at all.

As a matter of fact, we can just have a weird betting rule. For example:

You have to bet on the outcome of a fair coin toss. If you bet on Tails - this bet is repeated. What should be your betting odds?

We can deal with this kind of decision theory problem using regular probability space. Or, using weighted probability space. In the latter case, even though the coin is fair, we have

$P^{\prime }\left(Heads\right)=1/3$

Which, as we remember, is a completely normal situation, as far as values of weighted probability functions go.

Betting Application

As weighted probability values can be different from probability values and do not follow Conservation of Expected Evidence, they do not represent the credence that a rational agent should have in the event. Despite that, they can be still useful for betting. We just need to define an appropriate weighted utility function:

$U^{\prime }\left(X\right)=P\left(X\right)U\left(X\right)/P^{\prime }\left(X\right)$

Such a weighted utility may have weird properties, inherited from the probability function - like changing its values based on the evidence received. But as long as you keep using them in pair with weighted probability, they will be producing the same betting odds as regular utility and probability.

$P^{\prime }\left(X\right)U^{\prime }\left(X\right)=P\left(X\right)U\left(X\right)$

Weighted Probabilities in Sleeping Beauty

Now when we define weighted probability and understand its properties, we can see that this is what thirdism has been talking about the whole time.

Previously we were confused why thirder's measure for the coin being Heads shifts from 1/2 to 1/3 and back from Sunday to Wednesday, despite receiving no evidence about the state of the coin. But now it all fits into place. 

On Sunday we have a trivial case where probability space equals weighted probability space, there is nothing that can happen twice based on the state of the coin:

$P\left(Heads|Sunday\right)=P^{\prime }\left(Heads|Sunday\right)=1/2$

And likewise on Wednesday.

$P\left(Heads|Wednesday\right)=P^{\prime }\left(Heads|Wednesday\right)=1/2$

But during Monday/Tuesday there may be two awakenings by which we can weight the probability. Indexical sample space is different from regular sample space and therefore:

$P\left(Heads|MondayorTuesday\right)=P\left(Heads|Awake\right)=1/2$

$P^{\prime }\left(Heads|MondayorTuesday\right)=P^{\prime }\left(Heads|Awake\right)=1/3$

It would be incorrect to claim that thirders generally find weighted probabilities more intuitive than regular ones. In most non-trivial cases thirders are still intuitively using regular probabilities. But specifically when dealing with "anthropic problems", for instance, the ones including memory loss, they switch to the framework of weighted probability, without noticing it.

The addition of amnesia doesn't change the statistical properties of the experiment, nor is it relevant for the definition of a weighted probability space, but it can make weighted probabilities feel more appropriate for our flawed intuitions, despite complete lack of mathematical justification.

Likewise, Lewisian halfism is also talking about weighted probabilities and confuses them with regular ones. It proposes a different way to define a weighted probability function, while keeping the same weighted utility and, therefore, experimentally produces wrong results.

It has the advantage of appealing towards the principle that the measure of a coin being Heads shouldn't change without receiving new evidence. But it's a principle for regular probabilities, not weighted ones. The latter can be affected not only by received evidence but also by changes in the number of possible indexical events. 

So, as soon as we cleared the confusion and properly understood that we are talking about weighted probabilities, we can agree that Lewis's model is wrong, while Elga's model is right. In this sense thirders were correct all this time. All the arguments in favor of thirdism compared to Lewisian halfism stay true, they are simply not about probability.

Dissolving the Semantic Disagreement

So, with that in mind, let's properly dissolve the last disagreement.

As we remember, it is about the way to factorize expected utility and now we can express it like this:

$E\left(X\right)=P\left(X\right)U\left(X\right)=P^{\prime }\left(X\right)U^{\prime }\left(X\right)$

And we can see that this disagreement is purely semantic. According to the correct model:

$P\left(Heads|Awake\right)=1/2$

While according to thirdism:

$P^{\prime }\left(Heads|Awake\right)=1/3$

But these statements mean exactly the same thing. One necessary implies the other.

$P\left(Heads|Awake\right)=1/2\leftrightarrow P^{\prime }\left(Heads|Awake\right)=1/3$

As soon as we understand that one model is talking about probability while the other about weighted probability, the appearance of disagreement is no more. We have a direct way to translate from thirder language to halfer and back, fully preserving the meaning.

If only we had a time travel machine so that we could introduce the notion of weighted probability before David Lewis came up with "centred possible worlds" and "attitudes de se" and created all this confusion. In this less embarrassing timeline, when told about the Sleeping Beauty problem, people would immediately see that Beauty's probability that the coin is Heads conditionally on the awakening in the experiment is 1/2, while her weighted probability is 1/3. They would likely not even understand what is there to argue about.

Hopefully, our own timeline is not doomed to keep this confusion in perpetuity. It took quite some effort to cut through the decades long argument, but now, finally, we are done. In the next post we will discuss some of the consequences that follow from the Sleeping Beauty problem and develop a general principle to deal with probability theory problems involving memory loss.