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April 17, 2008

Cogitative Biases

Eliezer Yudkowsky’s article on heuristics and biases Cognitive biases potentially affecting judgment of global risks is one of my favorite on the topic.  I have read it a few times now and find it valuable every time.

Why is this topic interesting? Partly, because it is one of a handful of research areas in which sociology/psychology seems to contribute to fairly direct knowledge of how the brain works. More practically, H & B are interesting because one cannot avoid them.

Even when we are thinking hard and focusing narrowly on self awareness, it is practically impossible to avoid making these errors.  The best we can hope for is to leave notes for ourselves (sometimes it might be wise to leave actual physical notes!) instructing us to distrust our convictions. But knowing you are wrong because you trust your passed-self’s authority isn’t really avoiding thinking with H & Bs, it is merely switching the train onto a different set of rails. 

It seems true that, while many have explained the patterns of failure that can result from trusting these heuristics and biases, there are also important and powerful benefits from our brains having evolved to work this way.

In the table below, I outline simplistic thoughts trying to illustrate that every bias identified in Yudkowsky’s paper as a potential failure pattern also accounts for a great deal of human success.  Together, they not only represent potential pitfalls in perception and reasoning, but constitute a great deal of what we consider the basic, innate human ability to act, adapt and get on in the World.

Cognitive Bias

Heuristic Strength

Availability—Information at-hand is adequate for risk assessment

Reasonable good caching algorithm in that recent work is most likely to be accessed next; Coherence of action (focus) is maintained

Hindsight—Story explaining past accounts for outcome

Learning! Emotional conviction of the connection between cause and effect. Coherent stories stick in our memories very well.

Black Swans—Assume unlikely events are more unlikely than they often are (This is like Availability in that current or average are assumed to continue)

Allows action in an uncertain world. Assume continuity because planning for all unlikely events uses up all your resources.

Conjunction

Great first guess because local action is most likely (action-at-a-distance is a bad first guess—and spooky)

Probability Increases with Detail (A mixing of Conjunction and Availability)

Personal experience works this way in that we always have most detail about personal experience—and personal experience is taken to be most trustworthy.

Confirmation—Confirming Facts are Selected from Sea of Facts

Intentions and Synchronicity

Anchoring, Adjustment and Contamination (Availability & Conjunction)

Data we experience in succession are often related

Affect

Action in uncertain situations is better than being frozen. Rules available when little info is available but you an answer is still required.

Scope Neglect

Rules available when little info is available but you an answer is still required.

Calibration and Over Confidence

Rules available when little info is available but you an answer is still required.

Bystander Apathy

Optimization and queuing heuristic.

April 13, 2008

Fruita Colorado Mountain Biking Trip

My 14er-climbing buddy and I went to Fruita, Colorado for a weekend of Mountain biking single tracks.  It was awesome!  We did two rides near Fruita, one in the Kokopelli Trails area and one in the Bookcliffs.  We rode about 25 miles of single track in all with great views of the river a few challenging climbs and some really fun descents (Shoots and Ladders).

The best guide books we found were the "Latitude 40 Maps: Fruita Grand Junction Recreation Topo Map" and the "Fruita Fat Tire Guidebook" by Troy Rarick and Anne Keller.  Both are available at a great bike shop downtown Fruita--Over The Edge.

Below are some random selections from my Fruita Flickr pool:

www.flickr.com
Dr. Skippy's items tagged with Fruita More of Dr. Skippy's stuff tagged with Fruita

April 10, 2008

Statistics of Coin-Toss Patterns II

This is a follow-up to my first coint toss post a few days ago.  I was surprised at the size of the difference between the average tosses for various patterns of the same length.  For example, for the patterns THT vs THH, the average number of coin tosses to achieve the patterns differed by 2 tosses.

In my last post, I argued that the difference can be understood by looking at the number of permutations of n coin tosses without the target pattern.  Below, I make this very explicit by calculating the fraction of possible coin toss permutations without the pattern over the total number of permutations.

For low numbers of coin tosses, the differences are small (the are equal up to 4 tosses for the THT and THH example; 5 for the example patterns HT and HH. In the calculation of average value of n, the terms are multiplied by \frac{1}{2^n}) The fraction of permutations of coin tosses without THT and THH are compared in the plot below.  The corresponding data is in the table below.

 

THT vs THH Permutations
Fraction of total coin toss sequecne permutations
without THT (red) and THH (blue).

 


I extended the code a little bit more to make this calculation.  In Python script available in previous last post, I used a designed counting scheme for the series sum calculation. In this case, I extended the class to count the permutations without the pattern without ensuring the last len(pattern) terms equal the pattern (i.e. I just counted permutations this time). You can get the update here.

Tosses Total Permulations Coin Toss Permutations WO Pattern (THT) fraction Coin Toss Permutations WO Pattern (THH) fraction
1 2 2 1.000 2 1.000
2 4 4 1.000 4 1.000
3 8 7 0.875 7 0.875
4 16 12 0.750 12 0.750
5 32 21 0.656 20 0.625
6 64 37 0.578 33 0.516
7 128 65 0.508 54 0.422
8 256 114 0.445 88 0.344
9 512 200 0.391 143 0.279
10 1,024 351 0.343 232 0.227
11 2,048 616 0.301 376 0.184
12 4,096 1,081 0.264 609 0.149
13 8,192 1,897 0.232 986 0.120
14 16,384 3,329 0.203 1,596 0.097
15 32,768 5,842 0.178 2,583 0.079
16 65,536 10,252 0.156 4,180 0.064
17 131,072 17,991 0.137 6,764 0.052
18 262,144 31,572 0.120 10,945 0.042
19 524,288 55,405 0.106 17,710 0.034
20 1,048,576 97,229 0.093 28,656 0.027
21 2,097,152 170,625 0.081 46,367 0.022

Monty Hall, Monkeys and M&Ms (Link)

More on the topic of probabilities and statistics...

This article was interesting to me because it deals with both a classic probability brain-twister (the Monty Hall problem) and cognitive dissonance (I am working on a post on heuristics and biases). Check out this NYT article regarding a potential fundamental Monte Hall error in some experiments supporting cognitive dissonance.

April 03, 2008

Statistics of Coin-Toss Patterns

Yesterday, I watched Peter Donnelly's TED presentation on statistical mind-benders. Among other things (statistician jokes!), Peter observes that humans don't have good intuition for some kinds of statistical thinking. In the presentation, Donelly posses a coin toss problem to demonstrate his point.  He chooses one that is easy to get wrong.

Consider a series of fair coin tosses. For example, one possible sequence of coin tosses is THTTHHTHTTH. How many tosses are required to get a particular pattern? How does this depend on the length of the pattern?

Peter poses a concrete question as follows.  Consider the pattern HTH. If we do the experiment of tossing a coin repeatedly and counting the number of tosses, we find that the first occurrence of HTH arises in some average number of coin tosses n_{HTH}. For a different pattern, say TTH, we can repeat the experiment and find that the first occurrence of this pattern arises in some average number of coin tosses n_{TTH}.

One of the following statements must be true:

(a) n_{HTH} = n_{TTH}
(b) n_{HTH} \gt n_{TTH}
(c) n_{HTH} \lt n_{TTH}

Which statement is correct?

My reflex reaction was (a). The heuristic leading to this conclusion is that the probability of getting TTH is the same as the probability of getting HTH in any 3 coin tosses, i.e.,

P(HTH)=P(TTH)=\frac{1}{8}.

On the other hand, if the pattern was HHH the probability of getting this pattern on any three coin tosses is the same. But intuitively, I expect to have to make more coin tosses on average to get this pattern.

It is difficult to get to the correct answer with this kind of reasoning. 

A little counter intuitively, (b) is the answer. It takes more tosses on average to get the pattern HTH than the pattern TTH. Peter spends some time arguing that this is plausible--watch his presentation to get those arguments. Below, I pursue calculating this for myself...

Remember that the average or expected value of n is calculated by,

=\sum{P(n)n}

The probability of any single coin toss sequence is \frac{1}{2^n} where n is the length of the sequence.

An approach to calculating P(n) is to count the number of permutations of coin-toss sequences of length 0 to n that contain the target pattern only in the last position. Each of these sequences has the length-dependent probability above, so that

P(n)=N(n-1)\frac{1}{2^{n-1}}(\frac{1}{2})=\frac{N(n-1)}{2^{n}}.

To take into account that some target patterns can overlap themselves, I force the last (l-1) places in the sequence to match the first (l-1) places of the pattern.  The chance of getting the pattern on the next toss is1/2.  So multiply by 1/2 to get the formula above.

The table below shows the function N(n-1) for the two patterns HTH and TTH. It is easy to see that for longer sequences, there are many more possible sequences not containing HTH than HHT. Since the mean is calculated as a sum of N(n-1) multiplied by the probability which only depends on the length of the sequence, HTH is going to have the greater expected n.

You can use a pad a paper to count sequences, but this gets out of hand quickly. I wrote a Python script to help keep things sorted out. This script makes all the calculations shown for the rest of this entry and can be downloaded here.

 

TossesCoin Toss Sequence
Permutations W/O
Pattern (TTH)
Coin Toss Sequence
Permutations W/O
Pattern (HTH)
422
543
675
7129
82016
93328
105449
118886
12143151
13232265
14376465
15609816
169861432
1715962513
1825834410
1941807739
20676413581
211094523833
221771041824
232865673396
2446367128801
2575024226030

 

It is easy to use these counts and the formula above to calculate the expected value of n for short patterns. Here is an example for a pattern of length 2. This calculation applies to either TH or HT and shows that the expected value of n is 4.00.

Coin
Tosses
Coin Toss Sequences
without Pattern (TH or HT)
Average Coin Tosses
(successive approximations)
321.250
432.000
542.625
653.094
763.422
873.641
983.781
1093.869
11103.923
12113.955
13123.974
14133.985
15143.992
16153.995
17163.997
18173.999
19183.999
20194.000

This series converges slowly and the numbre of permutations becomes very large for longer patterns. So, although this brute-force summing is straight forward, it runs up against practical limits surprisingly quickly.

Another approach to calculated the average n for a pattern is to use a Monte Carlo simulation of the coin toss events. The Python script above also performs this calculation. The table below shows the results for patterns up to length 4.

Pattern(Est.) E
Patterns of length 1
T2.00
H1.98
Patterns of length 2
TT5.99
HH6.00
HT4.02
TH4.01
Patterns of length 3
TTT13.95
HHH14.07
HTT8.03
THH8.02
THT9.95
HTH10.04
HHT8.00
TTH8.06
Patterns of length 4
TTTT30.07
HHHH30.15
HTTT16.10
THHH16.07
THTT17.96
HTHH17.98
HHTT15.92
TTHH16.16
TTHT18.09
HHTH18.00
HTHT20.14
THTH19.94
THHT18.12
HTTH18.20
HHHT16.13
TTTH16.0

This table is useful to compare the different results for various patterns of the same length.

Finally, the Monte Carlo method gives an estimate of the distribution of n for the patterns. (The numbers above are based on 20,000 sequences.) The histogram below shows the Distribution of Sequence length for the pattern TTTH.  The average sequence length is estimated to be 16 (from the table above). Because the distribution is asymmetric and has a very long tail to the right, the average doesn't appear near any interesting features (the peak, for example) of the distribution.

TTTH Distribution (20000 Trials) 

April 02, 2008

Words mean things--er, sort of

"It's just semantics" and "we're saying the same thing" are two responses to attempts at working out subtle and difficult differences I hate to hear. I find them lazy and cowardly.  They neither reveal common ground nor do they move anyone toward generative understanding of diversity.

These responses do not build trust.  They are disrespectful in the way they discredit one party's perceptions of the issues by elevating the perspective of the other who sees how things "actually" are. They imply it is smarter to gloss over facts and complex relationships between ideas. These responses imply that if you disagree, you should keep quiet until you see it the right way.

Holding disagreement, respecting one's own lack of understanding of an idea and going deeper without breaking trust with the people in the conversation is essential to progress.

And I think this is what many well meaning people who try to calm a contentious conversation with theses statements are trying to accomplish. But glossing over lanugage, distorting meanings, implying understanding where none exists, etc. doesn't work.

I was reminded of my experiences of colleagues killing a conversation by using the phrases above when I was reading Euphemism and American Violence by David Bromwich appearing in the New York Review of Books.  After reading the article, I realized there are many more and more subtle ways to accomplish the same thing.

Here is a representative quote from the article:

The "global war on terrorism" promotes a mood of comprehension in the absence of perceived particulars, and that is a mood in which euphemisms may comfortably take shelter. There is (many commentators have pointed out) something nonsensical in the idea of waging war on a technique or method, and terrorism was a method employed by many groups over many centuries before al-Qaeda—the Tamil Tigers, the IRA, the Irgun, to stick to recent times. But the "war on crime" and "war on drugs" probably helped to render the initial absurdity of the name to some degree normal. This was an incidental weakness, in any case. The assurance and the unspecifying grandiosity of the global war on terrorism were the traits most desired in such a slogan.

It is a fairly long article and well worth the read. Thanks to Chris for the pointer.


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