In tonight's post, I want to return to the Sleeping Beauty problem I discussed in the last two essays. This topic may seem a little dry but it may interest people who watched the Veritasium video about it on Youtube or who took an interest in it after reading the previous posts. It may well be that my readers are more interested in my life than my take on puzzling epistemological problems but I don't intend to talk about my life at all in this post. It will be a shorter essay than the others I have written recently. There is possibly more that I could say about my life but I'll save that for a later post.
First, I wish to discuss some mistakes I made in the last two posts. In the first of the two, "Probability, Time, and Bad Science." I described a thought experiment involving a pack of cards and four people and showed how, depending on the amount of information each person has, each will make a different estimate concerning the probability that the card on top is the Queen of Hearts. My more careful readers will have noticed that, in presenting this scenario, the four people can't be talking about the same pack of cards. This was not a mistake I made when I originally presented this thought experiment in the two posts "Probability and Schrodinger's Car" and "Probability and Schrodinger's Cat Part 2" I'll present the example again now, amended so that it makes more sense, so that it is obvious that the four people are all dealing with the same pack of cards. It runs as follows. We have a recently shuffled pack of cards. Person A assesses the probability of the top card being the Queen of Hearts as 1/52. Person B, who knows that the top 13 cards are all Hearts, assesses the probability as being 1/13. Person C knows that the top three cards are all Hearts and all Face cards and so assesses the probability as being 1/3. Person D knows that the top card is definitely the Queen of Hearts and so assesses the probability as being 1. Of course, if the Queen of Hearts is somewhere else in the pack, Person A's estimate remains good, justifiable, but at least one of the other three will come up with alternative estimates, depending on the information each has.
It is because a probability estimate is made by a particular person based on the information he or she has that I describe it as subjective. The word 'subjective' has many meanings but commonly if you say an opinion is subjective you mean that it is not based in reality. This is not the meaning I intend by the word 'subjective'. Each of the four people in the example above is making a rational judgement, based on the information each has, about the probability that the Queen of Hearts is at the top of the deck. What I mean in using the term 'subjective' is not that the estimate is irrational, based solely on opinions or prejudices - nor am I using the word subjective to say that their estimates are somehow illusory. Each of the four estimates is rationally justifiable given the amount of information he or she has. I do understand that this perhaps slightly idiosyncratic use of the word 'subjective' may cause confusion because the terms 'objective' and 'subjective' have different meanings to different people in different contexts. We commonly say that a scientist or psychiatrist is 'objective' when that person is considered unbiased – but the scientist or psychiatrists is still observing the world from a particular subjective position. It is this second sense that I mean by the word 'subjective'.
In the previous post I misused the word 'factor'. This was a stupid error that I regret. If something is a factor, it has a causal effect on something else. I used the word factor when discussing a partial causal link between obesity and heart attacks. The term I should have used was, perhaps, 'data points'. There is a correlation between the data points associated with obesity and the data points associated with heart attacks, although one neither necessarily nor sufficiently entails the other. We can say that obesity is a risk-factor for heart attacks but I should not have used the word 'factor' to describe heart attacks themselves.
In discussing the science behind schizophrenia, I slightly misrepresented the psychiatric position. These misrepresentations were not exactly mistakes – rather, I was presenting psychiatry in a better light than it deserves. In the first of the two previous posts, I cited Wikipedia in saying that in some jurisdictions a psychotic episode has to last at least a month, and in other jurisdictions at least six months for a diagnosis of schizophrenia to be made. In fact, Wikipedia says, instead, that symptoms have to be present for at least a month (in some countries) and at least six months (in others). The Wikipedia entry does not mention psychotic episodes at all. As I said in the previous post, the failure of psychiatrists and the general public to recognise that schizophrenia, if it exists at all, is an episodic condition rather than a chronic condition is a major, unconscionable defect of psychiatric discourse, a failing reproduced in the Wikipedia article.
A second misrepresentation concerns 'risk factors'. I relayed the apparent facts that research has suggested that a difficult childbirth is a risk-fact associated with schizophrenia and that cannabis use is a risk-factor associated with schizophrenia. Of course, this research may be bogus. Even if this research is replicable, it misrepresents the psychiatric position. Many psychiatrists and the public generally tend to think of schizophrenia as a single thing with a single cause. The schizophrenia gene, for instance, was advanced in the past as the necessary and sufficient cause of schizophrenia, and currently cannabis use is being pushed as the cause. The more nuanced view, that there are many possible risk-factors associated with schizophrenia, and the more recently proposed view that adverse experiences during childhood are a major risk-factor, means that each patient needs to be understood and treated differently. The idea that schizophrenia has a single cause which scientists have yet to establish only plays into the hands of corrupt psychiatrists.
I turn now to the Sleeping Beauty Problem. I still support the halver position and my argument in favour of it still stands good. The criticism of Adam Elga's argument that I made in the previous post also stands good. I realise now however that I misrepresented the thirder solution.
In "Probability, Time, and Bad Science", I suggested that we should imagine Sleeping Beauty as being part of a large experiment involving a thousand Sleeping Beauties. I implied that, even if this is not the case, Sleeping Beauty should imagine herself to be part of such an experiment when estimating her credence that the coin came up heads. She should apply the Principle of Indifference to Sleeping Beauties rather than awakenings. We would expect 500 Sleeping Beauties to be involved in heads situations and 500 to be involved in tails situations. Yes, there are 1000 tails awakenings and only 500 heads awakenings but each tails awakening is only half as probable as a heads awakening. The probability that it is Monday and is or will be heads is 1/2, the probability that it is Monday and is or will be tails is 1/4, and the probability that it is Tuesday and is tails is 1/4.
Let us look at this large thought experiment from the thirder perspective. The a priori probability that the coin came up heads is 1/2. However, the thirder argues, Sleeping Beauty should assess here credence as 1/3. We have a contradiction. One way to resolve this contradiction would be to suppose that the a priori probability was always 1/3. This would mean that we should expect the coin to have come up heads in only 333 of the cases. But this would mean that there were 333 heads awakenings and 1332 tails awakenings. If this is the case, Sleeping Beauty should estimate the probability as being 333/1665 or 1/5. There is no way to make the a priori probability and Sleeping Beauty's credence agree. By my math, if the a priori probability is n, Sleeping Beauty's credence should always be n/(2-n).
This contradiction arises because we are trying to apply the Principle of Indifference to both Sleeping Beauties and to awakenings and we can't. In my view, we should apply the Principle of Indifference only to Sleeping Beauties. The thirder might argue that we should apply the Principle of Indifference only to awakenings. If so, the thirder could argue that all the awakenings are equally likely and that a Sleeping Beauty involved in a heads experiment is only half as probable as a Sleeping Beauty involved in a tails awakening. Yes, the Sleeping Beauties were evenly split between heads-results and tails-results – but not all Sleeping Beauties are equally probable and the credence of 1/3 stands. The problem with this is that the best possible way to define probability is to talk about the expected frequency of a result within a real or hypothetical set of possible result and we seem to be abandoning this principle. We are saying that, even though the probability of the coin coming up or having come up heads is 1/3, that 500 of the thousand Sleeping Beauties were involved in heads experiments. This just seems wrong to me. It just seems natural to me that we should apply the Principle of Indifference to Sleeping Beauties rather than to awakenings because the Sleeping Beauty experiments are all statistically independent of each other and the awakenings aren't.
We have discussed two thirder ways of explaining the large experiment and I have argued that both fail. There is a third option that I think best captures the thirder position. Yes, Sleeping Beauty assesses the probability of a coin coming up heads in a generic Sleeping Beauty experiments as 1/2. She may well say, "In any and all of the hypothetical Sleeping Beauty tests I am imagining, the coin has a fifty-fifty probability of coming up heads. This is true for all hypothetical Sleeping Beauties. In my specific case, as a non-hypothetical individual human being embodied in a particular situation and feeling myself to be the centre of the universe, it seems most rational for me myself to assess the probability as being 1/3." This is more or less the conclusion Adam Elga reaches, that a person should take spatio-temporal position into account when assessing probabilities. This is the strongest version of the thirder position. And if this is the version that thirders generally believe, there is no rational way to settle the dispute between halvers and thirders. Personally, I prefer the halver position because I think we should treat such problems abstractly, that we should take subjectivity (in the common sense of this word) out of the picture. To use the word "subjective" in the way I have been deploying it in the last couple of essays, both the halver and thirder position are subjective, but the former seems to me more rational than the later. There is, however, no way to adjudicate between the two positions. Adam Elga tries to prove the thirder position in his essay but, as I showed in the previous post, his argument fails.
This is all I have to say in this post. I may talk about my life in a subsequent post but I am not sure what is left to say. Because I am back at university again, it is likely that this blog will have to take a backseat to my studies. If people are still interested, send me a sign somehow and I will continue writing.
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