July, 2013


31
Jul 13

A probability cookbook

Randomness – Probability = Chance

Chance – Randomness = Fate

Fate + God = Predestination

Probability + Epistemology = Types of Randomness

Subjective Probability = Betting + Coherence

Propensity theory = Probability + Animism

Kolmogorov Axioms = Probability – randomness – chance

Probability + Complexity = Cryptography

Chaos + Ignorance = Randomness

Regression: Data = Signal + Noise

Bayesian:
Posterior = Prior  \times Likelihood
Prior + Data  \rightarrow Probability

Probabilitst:
Probability  \rightarrow Frequency

Statistical:
Frequency  \rightarrow Probability

Big Data:
Predictive value  \gg Model simplicity
High dimensions + Fast computers = De chao ordo


10
Jul 13

Updates to types of randomness

Just a quick note that I’ve gone through and made some revisions to A classification scheme for types of randomness. If you haven’t yet read this post, I’d highly recommend it. If you have, go read it again!


3
Jul 13

The hat trick

In his book Quantum Computing Since Democritus, Scott Aaronson poses the following question:

Suppose that you’re at a party where every guest is given a hat as they walk in. Each hat has either a pineapple or a watermelon on top, picked at random with equal probability. The guests don’t get to see the fruit on their own hats, but they can see all of the other hats. At no point in the evening can they communicate about what’s on their heads. At midnight, each person predicts the fruit on their own hat, simultaneously. If more than 50% of the guests get the correct answer, they’re given new Tesla cars. If less than 50% of the guests get it right, they’re given anxious goats to take care of. What strategy (if any) can they use to maximize their chances of winning the cars?

Answer: there is no strategy that works.

Kidding! Of course there’s a strategy, as you can tell by the length of this post. Did you come up with any ideas? At first glance, it seems like the problem has no solution. If you can’t communicate with the other party goers, how can you find out any information about the fruit on your own head? Since each person was independently given a pineapple or a watermelon with equal probability, what they have on their heads tells you nothing about what you have on your head, right?

My own initial strategy, after considerable (but not enough!) thought, was to bet on regression to the mean. Suppose you see 7 pineapples and 2 watermelons. The process of handing out hats is more likely to generate a pineapple/watermelon ratio of 7 to 3 than 8 to 2 (it’s most likely to generate an equal number of each type, with every step away from a 5/5 ratio less and less likely). Thus, I figured it would be best to vote that my own hat moved the group closer towards the mean. Following my strategy, we all ended up with goats. What did I do wrong?

The key to solving this problem is to realize that the initial process for handing out hats is irrelevant. All that matters is that, from the perspective of a given person, they are a random sampling of 1 from a distribution that is known to have either 7 pineapples and 3 watermelons, or 8 pineapples and 2 watermelons. Thus, each person knows that the probability a randomly sampled guest will have a pineapple on their head is somewhere between 70% and 80%. More precisely, it’s either 70% or 80%. In any case, so long as every person votes for themselves being in the majority, then the majority of guests will be voting that they are in the majority.

I simulated this strategy using parties of different sizes, all of them odd (to avoid the issue of having and equal number of each hat type). Here’s the plot, with each point representing the mean winning percentage with 500 trials for each group size. As always, you can find my code at the end of the post.

As you can tell from the chart, once we have 11 or more guests, it’s highly likely that we all win Teslas.

One way to look at this problem is through the lens of the anthropic principle. That is, we need to take into account how what we observe gives us information about ourselves, irrespective of the original process that made each of our hats what they are. What matters is that from the perspective of each party goer, their view comprises a random sampling from the particular, finite distribution of pineapples and watermelons that was set in stone once everyone had entered the room. In other words, even if the original probably of getting a pineapple was 99%, if you see more watermelons than pineapples, that’s what you should vote for.

This problem, by the way, is related to Condorcet’s Jury Theory (featured on the most recent episode of Erik Seligman’s Math Mutation podcast). Condorcet showed, using the properties of the binomial distribution, that if each juror has a better than 50% chance of voting in accordance with the true nature of the defendant, then the more jurors you add, the more likely the majority vote will be correct. And vice versa. Condorcet assumed independence, which we don’t have because our strategy ensures that every person will vote the same way, so long as the difference between types of hats is more than 2.

# Code by Matt Asher for StatisticsBlog.com
# Feel free to modify and redistribute, but please keep this header
set.seed(101)
iters = 500
numbPeople = seq(1, 41, 2)
wins = rep(0, length(numbPeople))
 
cntr = 1
for(n in numbPeople) {
	for(i in 1:iters) {
		goodGuesses = 0
		hats = sample(c(-1,1), n, replace = T)
		disc = sum(hats)
		for(h in 1:n) {
 
			personHas = hats[h]
			# Cast a vote based on what this person sees
			personSees = disc - personHas
 
			# In case of a tie, the person chooses randomly.
			if(personSees == 0) {
				personSees = sample(c(-1,1),1)
			}
 
			personBelievesHeHas = sign(personSees)
 
			if(personBelievesHeHas == personHas) {
				goodGuesses = goodGuesses + 1
				break
			}
 
		}
 
		if(goodGuesses > .5) {
 
			# We win the cars, wooo-hooo!
			wins[cntr] = wins[cntr] + 1
		}
	}
 
	cntr = cntr + 1
}
 
winningPercents = wins/iters
 
plot(numbPeople, winningPercents, col="blue", pch=20, xlab="Number of people", ylab="Probability that the majority votes correctly")

1
Jul 13

Morality needs probability, manifesto addendum

Just added to my Big Bright Green Manifesto Machine. You might need to read this through a couple times; it’s a difficult concept since it lives in a collective blind spot for us:

Doing ethics without probability is like performing surgery with a wooden spoon — it’s a blunt instrument capable of only the most basic operations, and more likely to kill the patient than heal them. Implicitly, we understand this need for probability in making ethical judgements, yet most people recoil when the calculus of probabilities is made explicit, because it seems cold, because the math frightens and confuses them, or because letting odds remain unestimated and unacknowledged allows people to confuse positive outcomes with moral behavior, sweeping hidden risks under the rug when things go well, or claiming ignorance when they don’t. It’s time to acknowledge — directly, explicitly, mathematically — that morality needs probability. For ethics to move forward it must be integrated with our knowledge of randomness and partial entailment.

Here’s an example of how we already take probability into account implicitly. If we retrieve our lost ball from someone’s yard without asking first, we justify this based on our belief that the owner is more likely to be bothered by us interrupting their dinner, than by our temporary trespass on their lawn. The greater the probability of great harm, the higher the level of certainty we demand. Our most heated debates involve situations where the probability of harm from both action and inaction is high. If someone’s dog is stuck in a hot car on a sunny day, should you break in and try to save it? Does the chance of a dog dying of heatstroke justify a forced entry that will probably result in expensive damage and an irate owner (though it’s possible they would be grateful instead). If you decide to break in, how long should you wait first? What prior distribution should you put on the owner’s return time, and how do you update your prior as time goes by? If the waiting time is chi-square on low degrees of freedom, your concern for the dog might be unjustified. If it follows the unreliable friend distribution, you may be that dog’s only hope.

As I hope is becoming clear, questions of morality cannot be resolved without asking questions about probability. If the example above seems trivial (perhaps the owner’s property rights trump your concern for a dog), then substitute the animal for a toddler who looks uncomfortably warm. Now how long do you wait, and how do you deal with the risk that smashing a window might harm the child?