I promised a few weeks back to write a few blogs on certain paradoxes I find interesting. Being the stats geek I am, especially when it comes to this blog, I'm going to kick off with the Sleeping Beauty Problem.
You can read the Wikipedia link above for all the gory details, but the setup is basically as follows:
Sleeping Beauty is involved in a strange experiment. She's put to sleep on Sunday using a special pill that not only knocks her out cold, but wipes out any memory of the previous day should she take it a second time.
She wakes up on Monday. A coin is tossed (without Sleeping Beauty seeing the toss or result) and if it is heads she is asked The Question, given her payoff and sent on her way.
If it comes up tails then she is asked The Question, given the pill again (so she won't remember Monday) and put back to sleep. The next morning she is asked The Question again, and then given her payoff and sent on her way.
Essentially, each time on waking, she has no idea if it is Monday or Tuesday.
The Question is this: "What is the probability the coin landed heads?"
What is the correct answer?
Now, the coin is completely fair, so arguably the probability the coin landed heads is 50%.
On the other hand, if the experiment is repeated time and time again (such as, in the Wikipedia example, 1000 times) then 500 times the coin is up heads and it is Monday, 500 times the coin is up tails and it is Monday, but an additional 500 times the coin is up tails and it is Tuesday. Then, the probability the coin landed heads is only 1/3.
Obviously there's a contradiction, and its a somewhat subtle one. For starters, I used the word 'probability' above. Wikipedia used the word 'credence' - which is a term for a particular interpretation of probability - the subjective Bayesian one. It's not the absolute objective probability, which undoubtedly is 1/2 for the coin.
As you might have picked up from my previous blogs involving Bayesian statistics, I'm very much a Bayesian. I will often use terms like 'probability' to mean my level of belief in something.
There's also the peculiar nature of the sampling here - it's a bit like me running a medical trial where I record twice any time someone gets better, and record once any time someone doesn't. It's a completely crazy way to record the results, and it means if I pick a result at random from my records I am more likely to get a positive result than I really should. That's perhaps the root cause of why the two answers are different.
I would recommend you go away and read the link above and related pages, and try to get to grips with the arguments on either side. The reason I find this interesting as a problem is that it really highlights how subjective the interpretation of probability can be. Whereas classically we might think probability is an objective value, we generally take that to be in some kind of fairly well informed and evenly sampled circumstance. In other words, the probability of an event is the proportion of how many times that event will happen in some sample of identical events.
This is very clean, but has difficulties. For one you disallow the power of Bayesian reasoning, and for another it becomes really hard to talk about unique events and whether they have probabilities at all - there's no population of identical events to measure them from.
For the Bayesian, the probability is the belief in an event. The value is not so much indicative of how many times the event will happen, but of how often I can get the answer to a question right. In this case things are skewed by how often I'm asked the question, but it can also be skewed by how well I understand the situation.
This paradox was once posed at a conference on Bayesian statistics I attended, and having already known of the problem for some time I phrased my answer broadly like this:
"The probability the coin landed heads is 0.5, but I'm always going to put money on it coming down tails."
I think that's about as good an answer to the problem as any.
