Posted July 1, 2013 by Dr. Henri Montandon in brain experiments
 
 

What chance do we have of understanding your thought, Reverend Bayes?

 

Your friends and colleagues are talking about something called “Bayes’ Theorem” or “Bayes’ Rule”, or something called Bayesian reasoning.  They sound really enthusiastic about it, too, so you google and find a webpage about Bayes’ Theorem and…

 

http://yudkowsky.net/rational/bayes

From the site:

   While there are a few existing online explanations of Bayes’ Theorem, my experience with trying to introduce people to Bayesian reasoning is that the existing online explanations are too abstract.  Bayesian reasoning is very counterintuitive.  People do not employ Bayesian reasoning intuitively, find it very difficult to learn Bayesian reasoning when tutored, and rapidly forget Bayesian methods once the tutoring is over.  This holds equally true for novice students and highly trained professionals in a field.  Bayesian reasoning is apparently one of those things which, like quantum mechanics or the Wason Selection Test, is inherently difficult for humans to grasp with our built-in mental faculties.

   Or so they claim.  Here you will find an attempt to offer an intuitive explanation of Bayesian reasoning – an excruciatingly gentle introduction that invokes all the human ways of grasping numbers, from natural frequencies to spatial visualization.  The intent is to convey, not abstract rules for manipulating numbers, but what the numbers mean, and why the rules are what they are (and cannot possibly be anything else).  When you are finished reading this page, you will see Bayesian problems in your dreams.

It is always fun for us to review a site which explains clearly what has been an intellectual puzzle seemingly forever, especially if it has a role in theories about the brain and consciousness. The following is a notorious puzzle launched into geekdom by Marilyn vos Savant in 1990:

   Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Ms. vos Savant was excoriated for her answer: it is in the contestant’s best interest to switch. An apocryphal story tells how even the mathematical genius Paul Erdös refused to believe this result, until it was demonstrated to him by a computer simulation. (You can try the simulation approach yourself at http://www.nytimes.com/2008/04/08/science/08monty.html?_r=0#).

We don’t really want you to worry about this particular problem very much. Just remember for now that it can be solved using Bayes; theorem. It is the kind of problem that interests us. Two things can be said about the path to a solution: 1. The solution involves probability; 2. The probability changes once we have more information. This kind of probability is termed conditional probability and its invention/discovery is attributed to Reverend Thomas Bayes, an 18th century English mathematician and Protestant minister. (Little is known about Bayes, including whether the picture at the beginning of this blog is actually him or not.)

What we do have for you is a limpid web site of great value. It is the best explanation of conditional probability we have found in several dozen other texts.

Depicted graphically, here is one example of an intuitive approach to a problem stated in the web site:

If a woman tests positive for breast cancer, what are the odds she actually has breast cancer?

100 women

Starting with an abstract depiction of 100 women…

1% have breast cancer

1 woman out of 100 (depicted by the red circle with the blue border) will have breast cancer

80% of the women who have breast cancer test positive for it.

80 out of 100 women who do have breast cancer will test positive for it, as depicted by the red circles with the green borders.

Approximately 10% (actually 9.5%) of women who test positive for breast cancer do not have it.

About 10% of all women tested will test positive for breast cancer, even though they do not have breast cancer. Getting back to the original question:

If a woman tests positive for breast cancer, what are the odds she actually has breast cancer?

By visualizing the data, we are trying to engage the ability of the brain to reason visually, i.e. to prime a visual intuition module.

An intuitive approach goes like this:

If you test 10,000 women, 100 will have breast cancer.

Of these 100, 80 will have positive mammographies.

Of the 10,000 women tested, 9900 will not have breast cancer, but 950 will get positive mammographies.

The total number of women with positive mammographies is 950 + 80 = 1030.

Of the 1030 women who test positive for cancer, only 80 will actually have cancer.

Therefore the percentage of women who tested positive who actually have cancer is 80/1030 = 7.8%.

If you can look at the diagrams above and then read the explanation,. and the answer really seems obvious, you have successfully engaged an intuitive cortical work-around for a kind of reasoning which is not present for access within most human brains.

Starting with an intuitive approach, the formalism of the full Bayesian analysis is presented step by step with worked examples for each concept, a lexic approach by means of which even I can learn Bayesian statistics.[i]


[i] The term lexic was coined by Dr. Baars and myself to describe what is the most powerful pedagogical method ever devised. If you are not familiar with it, check out publications from the Transnational College of Lex.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Dr. Henri Montandon