# Are human minds statistical machines?

Human minds are the mother of all interesting things since anything that we might consider interesting is so because our minds make us believe so. Seems then reasonable that all kind of philosophical issues and scientific problems cannot be properly addressed unless we correctly understand how our minds work, but what we know about how they work?

Cognitive Science offers many theories on how any mind might work, but when it comes to our minds there seem to be evidences put forward by psychologists that, whatever the way they work,  human minds do not abide to the laws of probabilities.

Several attempts have been made to explain these results, and one of the latest comes from the hand of Quantum Mechanics… No kidding.

So when I saw this valiant attempt from theoretical physicists to explain how the human mind works by using their all mighty and powerful Quantum Hammer, I thought it was a good moment to explain an alternative solution that I myself worked out long, long ago, after being exposed to this problem by philosopher Paul Thagard in his excellent book MIND.

Also, Sister Hot is my assistant and I need her to prove my point which is that our minds might abide to probability laws more than we think after all. If you want to know how she is going to assist me you need to keep reading; probability can be sexy 😉

# MIND

Long ago I took a course in Cognitive Science and we were asked to buy the excellent book MIND from philosopher Paul Thagard. The book explores several approaches that attempt to explain how the human mind works, and in the Chapter 2 (pag. 38) dedicated to Logic we can read:

Just as Johnson-Laird has challenged the relevance of formal logic to human deductive reasoning, psychologists have done experiments that suggests that human inductive  reasoning may not have much to do with probability theory.

As an example of evidences on why this might be so Paul Thagard continues:

Tversky and Kahneman (1983), for example, have shown that people sometimes violate the rule that the probability of a conjunction will also be less or equal to the probability of one of its conjuncts, P(p & q) ≤ P(p).

There are many examples to illustrate this, but I will give here a conspicuous one for clarity purposes. Imagine we ask people to compare probabilities, and we ask them about the probability of becoming a nun vs. the probability of becoming a nun AND a prostitute. Most people, even not religious ones, will say

P(becoming a Nun AND a Prostitute) < P(becoming a Nun)

And this is correct. Now imagine again we ask people about the probability of becoming a porn star vs. the probability of becoming a porn star AND a prostitute. Now, experiments seems to show that most people will say

P(becoming a Porn Star AND a Prostitute) > P(becoming a Porn Star)

And probability theory say this is wrong. So what is going on here? Do we need Quantum Mechanics, or even more complex formulations, to explain this phenomenon? I don’t really think so. This seems to me a clear case showing that for those that only have a hammer everything looks like a nail, but I don’t really blame the theoretical physicists for using their fancy hammer since we all are guilty of having our personal toolbox to deal with problems, and just like theoretical physicists have their toolbox, so do I have mine.

# My Occam’s Hammer

I once read there was this French politician whose name I don’t remember (or maybe I never knew) who said something like “I am against referendums because French people never answer to the questions they are asked to”, and I believe this is exactly what is going on here.

When we are asked about P(becoming a Porn Star) our minds actually respond to the question P(becoming JUST a  Porn Star), so what we are actually saying is:

P(Porn Star AND Prostitute) > P( Porn Star AND NOT a Prostitute)

And guess what… This is statistically absolutely TRUE! So there, that’s it, no Quantum Probability needed. If this is the reason for human bias when given these statistically unsound answers, the experiments published by physiologists would prove humans guilty of misunderstanding questions, but they say nothing against our abilities to asses probabilities.

I can understand why physicists would overlook this cognitive bias but I am surprised that psychologists did too. Maybe once you have an interesting result you are too excited to fool prove it? Maybe I am doing that myself right now? But I truly believe my explanation is way more reasonable than going Quantum and, among equally plausible explanations, I just go Occam.

Now,  this alternative explanation is not proof that our minds are mathematically sound statistical machines since, unfortunately, we are subject to a whole fauna of cognitive biases and a number of these might lead us into wrongly estimating the probabilities of an event, but it shows that before going Nuclear (Physics) on a problem maybe we want to try some homey common sense.

# Update

Mr. Corey kindly pointed me to this post in lesswrong where it seems that Tversky and Kahneman knock down the interpretation I gave above about their results with another experiment which tries to make sure students interpreting the experimental instructions in an unexpected way is not the reason for the bias. Let’s see if that is truly so, this is the experiment:

Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequences of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win \$25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet.

1. RGRRR
2. GRGRRR
3. GRRRRR

The question never mentions the word “probable” so that students construct whatever solution they want without having to think what the researchers mean by “probable”. Okay, fair enough.

It turns out that 65% of the students gave the wrong answer (option two). So, do I need to eat my words and go Quantum? Or maybe is it possible that a tiny change in how the question is interpreted alters the results? Well, let’s do a tiny change and see what happens:

Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the set of greens (G) and reds (R) will be recorded. You are asked to select one set, from a set of three, and you will win \$25 if the set you chose appears on successive rolls of the die. Please check the set of greens and reds on which you prefer to bet.

1. RGRRR
2. GRGRRR
3. GRRRRR

So now we have one set of five tosses and two sets of six tosses, let’s calculate which set among these is more likely to appear:

1. $P(1G 4R) = \binom{5}{1}(2/3)(1-2/3)^4 = 0.041$
2. $P(2G 4R) = \binom{6}{2}(2/3)^2(1-2/3)^4 = 0.082$
3. $P(1G 5R)=\binom{6}{1}(2/3)(1-2/3)^5 = 0.016$

Oh well, look at that, now the option chosen by 65% of people is twice as likely  as the “right” one! So what now? Do we go Quantum? I don’t think so, we still go Occam.

All these experiments trying to prove we humans suck big time at assessing probabilities introduce “poisoned candies” to lure the subjects of the experiment into the wrong answer, but what are they exactly proving? That we can be fooled? A skilled magician can turn the smartest person in the world into a sucker using all kind of deception tricks.

What psychologists are doing reminds me to those fortune tellers that they do not only fool their customers but themselves; their tricks bites them back. If we use “poisoned candies” in these experiments we are incurring in confirmation bias.

If psychologists truly want to prove we humans hugely miscalculate probabilities then they need to figure out experiments with no poisoned candies, that is, with options where no alternative probabilistic explanation makes right the “wrong” answer, but my feeling is that if they do so this huge effect they claim will disappear.

## 11 thoughts on “Are human minds statistical machines?”

1. 🙂 hey, welcome back. Nice!
On another note, answering questions at such a high and complex level of behavior such as nuns and prostitutes and so on, it is well worth remembering that we can’t even explain how we are able to talk creatively, recognize objects and so on. Compared to such “simple” questions, high level complex behavior, contingent on all kinds of diverse factors, is a bridge that is eons away, and for all we know we might never understand it…

• Hey Rameez!

Thank you very much for reading and still be around! 🙂 I really appreciate your input since I know you’re a brilliant man and it feels good when intellects like yours pay attention to one’s random thoughts.

Yes, you’re right, Cognitive science (the study of minds) is in its infancy and it is just a mixture of the very few things we know about this subject, which is virtually nothing.

And you are right again when you say we might never fully understand our minds since, the same way we cannot use a theorem to prove itself in mathematics, it is reasonable to think that we might not be able to prove that our mind can fully understand itself one day.

Still, Cognitive Science is really exciting. One of the most interesting subjects in the course were the intents to understand and model consciousness.

Though one of the most embarrassing moments in the course were the day we tried to model sense of humor and the professor challenged us to tell joke in class… I made the mistake to raise my hand and to tell a Spanish joke in the conservative state of Georgia (USA)… Not a good idea, believe me. 😀

• haha. Humor can be a difficult one. Just recently, two jokes of mine (in an academic setting) fell flat to awkward, uncomfortable silence 🙂
And yes, cognitive science is very exciting indeed. In fact, were it not for my laziness, I already should have written a post on a new exciting idea in the cognitive sciences, which is very much in its infancy.
(thanks for the kind words)

2. Entsophy says:

P(Porn Star AND Prostitute) > P( Porn Star AND NOT a Prostitute)

You nailed it. Most, but perhaps not all, of these these results come from the discrepancy between the questions being asked and the implicit question being answered.

The foundations of quantum mechanics are at least an order of magnitude less secure than classical statistics, so any appeal to the quantum is likely a waste of time.

Perhaps a better a better question for psychologists to ask is “how poorly can people’s statistical reasoning be before it starts to have serious consequences for themselves?”

• Ent! well, well, more talented people commenting… how do you like that!

Perhaps a better question for psychologists to ask is “how poorly can people’s statistical reasoning be before it starts to have serious consequences for themselves?”

I think that is a better way to put things, and I assume we humans cannot be that bad as the 85% fail rate the psychologists studies suggest since Darwin laws would have probably taken care of individuals failing to assess:

P( Lion Kills me | Get Close to Lion ) vs. P(Lion Kills me)

3. Corey says:

A clever interpretation — but one that Tversky and Kahneman knocked down in the very same 1983 paper that Thagard cited. Here’s the game K&T invented to address these kinds of objections, which was put to 125 undergrads:

Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequences of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win \$25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet.

1. RGRRR
2. GRGRRR
3. GRRRRR

“65% of the subjects chose sequence 2, which is most representative of the die, since the die is mostly green and sequence 2 contains the greatest proportion of green rolls. However, sequence 1 dominates sequence 2, because sequence 1 is strictly included in 2. 2 is 1 preceded by a G; that is, 2 is the conjunction of an initial G with 1. This clears up possible misunderstandings of “probability”, since the goal was simply to get the \$25.” http://lesswrong.com/lw/ji/conjunction_fallacy/

• Thank you very much Corey for your comment! I really appreciate it.

I was going to reply to you here but the answer was growing a bit to much and, since I think the experiment you mention is highly relevant to the point I made in my post, I have updated the post with my answer.

The conclusion is that, no, we still go Occam since questions can be easily misinterpreted and I show you how this could have happened in the experiment you mention.

• Corey says:

Since the challenge is very clear about what conditions will win the \$25, I think you haven’t knocked down the K&T argument so much as rephrased it. Your transition from sequences to sets has a lot of potential as a concrete model of the representativeness heuristic.

4. Corey,

Since the challenge is very clear about what conditions will win the \$25, I think you haven’t knocked down the K&T argument so much as rephrased it.

That is a lawyer argument; “the contract is clear!”… so what? here you have a nice collection of wrong answers to very clear questions where the subject lost a lot more than \$25.

What now? Do we explain this with Quantum Mechanics too? We should not underestimate the ability humans have to misunderstand questions o mis-construct answers.

Your transition from sequences to sets has a lot of potential as a concrete model of the representativeness heuristic.

What they call “representativeness” seems to me nothing more than P(Die | Sequence), that is, what is the most likely Die configuration given we have a given Sequence. Whereas the problem was asking P(Sequence | Die).

But if people were answering to that “representativeness” from K&T then it would still deepen my point which is that people misunderstand questions because, guess which of the three options have a closer “representativeness” / P(Die | Sequence) to the die described in the problem…. that’s right… option number two again.

But hey, it is actually very easy to prove me wrong, change option one from RGRRR to RGRG… (RGRG being better than GRGR to difficult the use of logic since GRGR is contained in option two) Now my argumentation would not hold since RGRG would be the most likely set, the most likely sequence, and also have the highest “representativeness” / P(Die | Sequence)

So, if we still get something even close to 65% of students choosing option number two, it will be time for me to eat my words and give Quantum a chance…

But I think we both know that is not going to happen, if K&T could get the same results with RGRG they would have publish it, but doing so means to take the poisoned candy away from the experiment… and with no candy, no results.

Note: We would still have the 4R2G relationship in option two that might lead astray some students by confusing 4G2R with 4R2G (I did by the way), so to eat my words I need that to be fixed too with option two showing something like GRGRGR or stating the probabilities for Red and Green in the problem instead giving the number of Green and Red sides.

5. nikosms says:

Apart my previous comment linking to another post by Fran related to this post, i’d like to point out simply the following:

When we are asked about P(becoming a Porn Star) our minds actually respond to the question P(becoming JUST a Porn Star), so what we are actually saying is:

P(Porn Star AND Prostitute) > P( Porn Star AND NOT a Prostitute)

And guess what… This is statistically absolutely TRUE!

Actualy this is NOT true. But it is plausible (and true in some cases).