Very good examples of perceptions driving self-selection.

It might be useful to discuss direct and indirect effects.

Suppose we want to compare fatality rates if everyone drove a Volvo versus if no one did. If the fatality rate was lower in the former scenario than in the latter, that would indicate that Volvo's (causally) decrease fatality rates.

It's possible that it is entirely through an indirect effect. For example, the decrease in the fatality rate might entirely be due to behavior changes (maybe when you get in a Volvo you think 'safety' and drive slower). On the DAG, we would have an arrow from volvo to behavior to fatality, and no arrow from volvo to fatality.

A total causal effect is much easier to estimate. We would need to assume ignorability (conditional independence of assignment given covariates). And even though safer drivers might tend to self-select into the Volvo group, it's never uniform. Safe drivers who select other vehicles would be given a lot of weight in the analysis. We would just have to have good, detailed data on predictors of driver safety.

Estimating direct and indirect effects is much harder. Typically it requires assuming ignorability of the intervention and the mediator(s). It also typically involves indexing counterfactuals with non-manipulable variables.

as an aside: a machine learning graduate student worked with me last year, and in most simulated data settings that we explored, logistic regression outperformed SVM

I'd like to ask those people who downvote this post for their reasons. I thought this is a reasonable antiprediction to the claims made regarding the value of a future galactic civilisation. Based on economic and scientific evidence it is reasonable to assume that the better part of the future, namely the the time from 10^20 to 10^100 years (and beyond) will be undesirable.

If you spend money and resources on the altruistic effort of trying to give birth to this imaginative galactic civilisation, why don't you take into account the more distant and much larger part of the future that lacks any resources to sustain given civilisation? You are deliberately causing suffering here by putting short-term interests over those of the bigger part of the future.

How about spreading rationality?

This site, I suspect, mostly attracts high IQ analytical types who would have significantly higher levels of rationality than most people, even if they had never stumbled upon LessWrong.

It would be great if the community could come up with a plan (and implement it) to reach a wider audience. When I've sent LW/OB links to people who don't seem to think much about these topics, they often react with one of several criticisms: the post was too hard to read (written at too high of a level); the author was too arrogant (which I think women particularly dislike); or the topic was too obscure.

Some have tried to reach a wider audience. Richard Dawkins seems to want to spread the good word. Yet, I think sometimes he's too condescending. Bill Maher took on religion in his movie Religulous, but again, I think he turned a lot of people off with his approach.

A lot has been written here about why people think what they think and what prevents people from changing their minds. Why not use that knowledge to come up with a plan to reach a wider audience. I think the marginal payoff could be large.

Consequences of non-consequentialism are disastrous. Just look at charity - instead of trying to get most good-per-buck people donate because this "make them a better person" or "is the right thing to do" - essentially throwing this all away.

If we got our act together, and did the most basic consequentialist thing of establishing monetary value per death and suffering prevented, the world would immediately become a far less sucky place to live than it is now.

This world is so filled with low hanging fruits we're not taking only because of backwards morality it's not even funny.

Perhaps a better title would be "Bayes' Theorem Illustrated (My Ways)"

In the first example you use shapes with colors of various sizes to illustrate the ideas visually. In the second example, you using plain rectangles of approximately the same size. If I was a visual learner, I don't know if your post would help me much.

I think you're on the right track in example one. You might want to use shapes that are easier to estimate the relative areas. It's hard to tell if one triangle is twice as big as another (as measured by area), but it's easier to do with rectangles of the same height (where you just vary the width). More importantly, I think it would help to show math with shapes. For example, I would suggest that figure 18 has P(door 2)= the orange triangle in figure 17 divided by the orange triangle plus the blue triangle from figure 17 (but where you show the division by shapes). When I teach, I sometimes do this with Venn diagrams (show division of chunks of circles and rectangles to illustrate conditional probability).

Wonderful. Are you aware of the Tuesday Boy problem? I think it could have been a more impressive second example.

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

(The intended interpretation is that I have two children, and at least one of them is a boy-born-on-a-Tuesday.)

I found it here: Magic numbers: A meeting of mathemagical tricksters

We should blame and stigmatize people for conditions where blame and stigma are the most useful methods for curing or preventing the condition, and we should allow patients to seek treatment whenever it is available and effective.

I think you said it better earlier when you talked about whether the reduction in incidence outweighs the pain caused by the tactic. For some conditions, if it wasn't for the stigma there would be little-to-nothing unpleasant about it (and we wouldn't need to talk about reducing incidence).

I agree with your general principle, but think it's unlikely that blame and stigma are ever the most useful methods. We should be careful to avoid the false dichotomy between the "stop eating like a pig" tactic and fat acceptance.

Sandy's husband is an asshole, who probably defends his asshole behavior by rationalizing that he's trying to help her. He's not really trying to help her (or if he is, he knows little about psychology (or women)).

Blame and judgment are such strong signaling devices that I think people rarely use it for the benefit of the one being judged. If it happens to be the best tactic for dealing with the problem, well, that would be a quite a coincidence.

--

I liked your post a lot, in case that wasn't clear. I think you are focusing on the right kinds of questions.

Yet one more variant. On my view it's structurally and hence statistically equivalent to Iterated Sleeping Beauty, and I present an argument that it is. This one has the advantage that it does not rely on any science fictional technology. I'm interested to see if anyone can find good reasons why it's not equivalent.

The Iterated Sleeping Beaty problem (ISB) is the original Standard Sleeping Beauty (SSB) problem repeated a large number N of times. People always seem to want to do this anyway with all the variations, to use the Law of Large Numbers to gain insight to what they should do in the single shot case.

The Setup

• As before, Sleeping Beauty is fully apprised of all the details ahead of time.
• The experiment is run for N consecutive days (N is a large number).
• At midnight 24 hours prior to the start of the experiment, a fair coin is tossed.
• On every subsequent night, if the coin shows Heads, it is tossed again; if it shows Tails, it is turned over to show Heads.

(This process is illustrated by a discrete-time Markov chain with transition matrix:

[1/2 1/2] = P
[ 1 0 ]

and the state vector is the row

x = [ Heads Tails ],

with consecutive state transitions computed as x * P^k

Each morning when Sleeping Beauty awakes, she is asked each of the following questions:

1. "What is your credence that the most recent coin toss landed Heads?"
2. "What is your credence that the coin was tossed last night?"
3. "What is your credence that the coin is showing Heads now?"

The first question is the equivalent of the question that is asked in the Standard Sleeping Beauty problem. The second question corresponds to the question "what is your credence that today is Monday?" (which should also be asked and analyzed in any treatment of the Standard Sleeping Beauty problem.)

Note: in this setup, 3) is different than 1) only because of the operation of turning the coin over instead of tossing it. This is just a perhaps too clever mechanism to count down the days (awakenings, actually) to the point when the coin should be tossed again. It may very well make a better example if we never touch the coin except to toss it, and use some other deterministic countdown mechanism to count repeated awakenings per coin toss. That allows easier generalization to the case where the number of days to awaken when Tails is greater than 2. It also makes 3) directly equivalent to the standard SB question, and also 1) and 3) have the same answers. You decide which mechanism is easier to grasp from a didactic point of view, and analyze that one.

• After that, Beauty goes on about her daily routine, takes no amnesia drugs, sedulously avoids all matter duplicators and transhuman uploaders, and otherwise lives a normal life, on one condition: she is not allowed to examine the coin or discover its state (or the countdown timer) until the experiment is over.

Analysis

• Q1: How should Beauty answer?
• Q2: How is this scenario similar in key respects to the SSB/ISB scenario?
• Q3: How does this scenario differ in key respects from the SSB/ISB scenario?
• Q4: How would those differences if any make a difference to how Beauty should answer?

My answers:

Q1: Her credence that the most recent coin toss landed Heads should be 1/3. Her credence that the coin was tossed last night should be 1/3. Her credence that the coin shows Heads should be 2/3. (Her credence that the coin shows Heads should be 1/3 if we never turn it over, only toss, and 1/K if the countdown timer counts K awakenings per Tail toss.)

Q2: Note that Beauty's epistemic state regarding the state of the coin, or whether it was tossed the previous midnight, is exactly the same on every morning, but without the use of drugs or other alien technology. She awakens and is asked the questions once every time the coin toss lands Heads, and twice every time it lands tails. In Standard Sleeping Beauty, her epistemic state is reset by the amnesia drugs. In this setup, her epistemic state never needs to be reset because it never changes, simply because she never receives any new information that could change it, including the knowledge of when the coin has been tossed to start a new cycle.

Q3: In ISB, a new experimental cycle is initiated at fixed times--Monday (or Sunday midnight). Here the start of a new "cycle" occurs with random timing. The question arises, does the difference in the speed of time passing make any difference to the moments of awakening when the question is asked? Changing labels from "Monday" and "Tuesday" to "First Day After Coin Toss" and "Second Day After Coin Toss" respectively makes no structural change to the operation of the process. Discrete-time Markov chains have no timing, they have only sequence.

In the standard ISB, there seems to be a natural unit of replication: the coin toss on Sunday night followed by whatever happens through the rest of the week. Here, that unit doesn't seem so prominent, though it still exists as a renewal point of the chain. In a recurrent Markov chain, the natural unit of replication seems to be the state transition. Picking a renewal point is also an option, but only as a matter of convenience of calculation; it doesn't change the analysis.

Q4: I don't see how. The events, and the processes which drive their occurence haven't changed that I can see, just our perspective in looking at them. What am I overlooking?

Iteration

I didn't tell you yet how N is determined and how the experiment is terminated. Frankly, I don't think it matters all that much as N gets large, but let's remove all ambiguity.

Case A: N is a fixed large number. The experiment is terminated on the first night on which the coin shows Heads, after the Nth night.

Case B: N is not fixed in advance, but is guaranteed to be larger than some other large fixed number N', such that the coin has been tossed at least N' times. Once N' tosses have been counted, the experiment is terminated on any following night on which the coin shows Heads, at the whim of the Lab Director.

Q5: If N (or N') is large enough, does the difference between Case A and B make a difference to Beauty's credence? (To help sharpen your answer, consider Case C: Beauty dies of natural causes before the experiment terminates.)

Note that in view of the discussion under Q3 above, we are picking some particular state in the transition diagram and thinking about recurrence to and from that state. We could pick any other state too, and the analysis wouldn't change in any significant way. It seems more informative (to me at any rate) to think of this as an ongoing prcess that converges to stable behavior at equilibrium.

Extra Credit:

This gets right to the heart of what a probability could mean, what things can count as probabilities, and why we care about Sleeping Beauty's credence.

Suppose Beauty is sent daily reports showing cumulative counts of the nightly heads/tails observations. The reports are sufficiently old as not to give any information about the current state of the coin or when it was last tossed. (E.g., the data in the report are from at least two coin tosses ago.) Therefore Beauty's epistemic state about the current state of the coin always remains in its initial/reset state, with the following exception. Discuss how Beauty could use this data to--

• corroborate that the coin is in fact fair as she has been told.
• update her credences, in case she accrues evidence that shows the coin is not fair.

For me this is the main attraction of this particular model of the Sleeping Beauty setup, so I'm very interested in any possible reasons why it's not equivalent.

I have the feeling you are talking about quite untypical NT people here (except maybe for example 3). Around me you would have defined "NT people" (even the term sounds strange to me) as being Aspies. That doesn't add up.

The thing that I have been most surprised by is how much NTs like symbols and gestures.

Here are some examples:

• Suppose you think your significant other should have a cake on his/her birthday. You are not good at baking. Aspie logic: "It's better to buy a cake from a bakery than to make it myself, since the better the cake tastes the happier they'll be." Of course, the correct answer is that the effort you put into it is what matters (to an NT).

• Suppose you are walking through a doorway and you are aware that there is someone about 20 feet behind you. Aspie logic: "If I hold the door for them they will feel obligated to speed up a little, so that I'm not waiting too long. That will just inconvenience them. Plus, it's not hard to open a door. Thus, it's better for them if I let the door close." To the NT, you are just inconsiderate.

• Suppose you are sending out invitations to a graduation party. You know that one of your close friends is going to be out of town that weekend. Aspie logic: "There is no reason to send them an invitation, since I already know they can't go. In fact, sending them an invitation might make them feel bad." If your friend is an NT, it's the wrong answer. They want to know they are wanted. Plus, it's always possible their travel plans will get canceled.

In each of these 3 examples the person with AS is actually being considerate, but would not appear that way to an NT.