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Paradoxes and Fallacies
Resolving some well-known puzzles with Bayesian networks
Stefan Conrady, [email protected]
Dr. Lionel Jouffe, [email protected]
September 3, 2013
Introduction to Bayesian Networks & BayesiaLab
Table of Contents
Introduction
Background & Objective
Notation
3
4
Paradoxes and Fallacies
Prosecutors Fallacy
5
Simpson’s Paradox
10
The Monty Hall Problem
16
Conclusion
20
Appendix
ii
Bayes’ Theorem
22
References
23
Contact Information
24
Bayesia USA
24
Bayesia Singapore Pte. Ltd.
24
Bayesia S.A.S.
24
Copyright
24
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Paradoxes and Fallacies
Introduction
Background & Objective
There are a number of paradoxes and fallacies that keep recurring as popular and mind-bending puzzles in
the media. Although there is (now) complete agreement among scientists on how to resolve them, the correct answers are often perplexing to the casual observer and still cause bewilderment.
We will start off with the fallacy of the transposed conditional, which has become rather infamous and is
better known as Prosecutor’s Fallacy. As the name implies, it is a problem often encountered in courts of
law, and there are numerous cases of incorrect convictions as a result of this fallacy.
No less serious are the potential consequences of Simpson’s Paradox, for instance, when determining the
treatment effect of a new drug under study. The effect of a drug on two subgroups may appear as the complete opposite of the treatment effect on the whole group.
On a much lighter note, the Monty Hall Problem has its origin in a television game show and might perhaps
be the most difficult puzzle to comprehend intuitively, even when explicit proof is provided. Respected
mathematicians and statisticians have struggled with this problem, and some of them have boldly proclaimed wrong solutions.
The counterintuitive nature of these probabilistic problems relates to the cognitive limits of human inference. More specifically, we are dealing with the problem of updating beliefs given new evidence, i.e. carrying
out inference. This cognitive challenge may seem surprising, given that humans are exceptionally gifted in
discovering causal structures in their everyday environment. Discovering causality in the world is quite literally child’s play, as babies start understanding the world through a combination of observation and experimentation. Our human intuition is actually quite good when it comes to reasoning from cause to effect; our
qualitative perception of such relationships (even under uncertainty) is often compatible with formal computations.
However, when it comes to reasoning under uncertainty in the opposite direction, from effect to cause, i.e.
diagnosis, or when combining multiple pieces of evidence, conventional wisdom frequently fails catastrophically. Even worse, the correct inference in such situations is often completely counterintuitive to people and
feels utterly wrong to them. It is not an exaggeration to say that their sense of reason is violated.
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Paradoxes and Fallacies
For more traditional computations, such as arithmetics, we have many tools that help us address our mental
shortcomings. For instance, we can use paper and pencil to add 9,263,891 and 1,421,602 as most of us
can’t do this in our heads. Alternatively, we can use a spreadsheet for this computation. In any case, it will
not surprise us that the sum of those two numbers is a little over 10.5 million. The computed result is entirely consistent with our intuition.
As this paper will show, the formally correct solutions of these probabilistic paradoxes are counterintuitive.
In addition to being counterintuitive, there are few tools assisting us in solving them. There is no spreadsheet that allows us to simply plug in the numbers to calculate the result.
Although we won’t be able to overcome inherent mental biases and cognitive limitations, we can now provide a very practical new tool for the correct inference in the form of Bayesian networks. Bayesian networks
derive their name from Reverend Thomas Bayes, who, in the middle of the 18th century, first stated the rule
for computing inverse probabilities.
Bayesian networks offer a framework that allows applying Bayes’ Rule for updating beliefs in the same way
spreadsheets are very convenient for applying arithmetic operations to many numbers. We will show how
restating these vexing problem domains as simple Bayesian networks offers near-instant solutions. Just as
spreadsheets help us perform arithmetic operations externally, i.e. outside our head, Bayesian networks offer a reliable structure to precisely perform inferential computations, which we can’t manage in our minds.
The visual nature of Bayesian networks furthermore helps (at least a little) in making these paradoxes more
intuitive to our own human way of thinking.
Beyond utilizing Bayesian networks as the framework, we will use BayesiaLab as the software tool for network creation, editing and inference. This allows us to leverage all the theoretical benefits of Bayesian networks for practical use via an intuitive graphical user interface.
Notation
To clearly distinguish between natural language, software-specific functions and example-specific variable
names, the following notation is used:
• BayesiaLab-specific functions, keywords, commands, etc., are capitalized and shown in bold type.
• Names of attributes, variables, nodes and are italicized.
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Paradoxes and Fallacies
Paradoxes and Fallacies
Prosecutors Fallacy
Crime dramas and live courtroom reporting have familiarized all of us with this situation, whether hypothetical or real: the prosecutor calls an expert to the witness stand and queries him about the reliability of
evidence found at a crime scene. The expert, typically a physician or medical examiner, will state something
like, “the probability of finding — by chance — the blood type at the crime scene which matches the one of
the defendant is about one in 1,000.” The prosecutor will presumably be satisfied with this answer and
probably paraphrase it in his closing argument to jury: “as you can see, the there is only a one in 1,000
chance that the defendant is innocent, and, therefore, it is clear beyond any reasonable doubt that the defendant is guilty.”
It wouldn’t be the Prosecutors Fallacy, if there wasn’t a problem with this seemingly plausible conclusion.
So, what’s wrong? Let us restate the expert witness’ testimony and furthermore clarify some implicit assumptions:
“The probability of identifying (or matching) some innocent person’s blood type at a crime scene by chance
(or sheer coincidence) is one in 1,000.” This is equivalent to the following:
P(Match=true∣Crime=false) =
1
= 10 −3
1, 000
In words, given that someone has not committed the crime, there is a 1/1,000 chance of identifying his or
her blood type at the scene of the crime by sheer coincidence.
However, the prosecutor claimed something else: “Given the evidence, there is only a 1/1,000 chance that
the defendant is not guilty,” which is a different statement:
P(Crime=false∣Match=true) =
1
= 10 −3
1, 000
So, should the jury find the defendant guilty? Maybe. Further assumptions are required to compute the correct probability of the defendant having committed the crime.
The first assumption is about the probability of a blood type match, given that one has actually committed
the crime. Let us assume that this probability is 1, i.e.
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P(Match=true∣Crime=true) = 1
Furthermore, we need to understand the base rate of the crime. For instance, statistics might tell us that this
crime happens only very rarely, e.g. only once in a city of 10,000 in a given time period. So, this is the marginal probability of being guilty:
P(Crime=true) =
1
= 10 −4
10, 000
Without any other knowledge, the probability of anyone in this city being guilty of such a crime is one in
10,000.
We can now use Bayes’ Rule1 to compute the probability in question, i.e. the probability of the defendant
being guilty. For a more compact representation, we will write:
“Match=true” = “Evidence”= “E”
“Crime=true” = “Guilty” = “G”
“Match=false” = “Not Evidence” = “¬E”
“Crime=false” = “Not Guilty” = “¬Guilty” = “¬G”.
The Bayes’ Rule will thus say:
P(G | E) =
P(E | G)P(G)
P(E)
The only unknown in this formula is P(E), i.e. the marginal probability of finding evidence by chance. To be
more precise, we can employ the law of total probability, which in our case translates into:
P(E) = P(E, ¬G) + P(E,G) =
P(E∣¬G)P(¬G)+P(E∣G)P(G)
We already know that
P(¬G) = 1 − P(G) ,
and hence we can compute:
1
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More details about Bayes’ Rule are provided in the appendix.
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Paradoxes and Fallacies
P(G | E) =
P(E | G)P(G)
P(E | G)P(G)
1⋅10 −4
1
=
= −3
≈
= 0.091 = 9%
−4
−4
P(E)
P(E | ¬G)P(¬G) + P(E | G)P(G) 10 ⋅ (1 − 10 ) + 1⋅10
10 + 1
So, given the evidence of a blood type match, the defendant has a 9% probability of being guilty, which is
presumably not enough for a conviction. However, as the marginal probability of being guilty is only
0.01%, the probability of the defendant’s guilt has risen 900-fold, given that the blood type matches.
However, the above approach may still prove to be cumbersome for practical use, especially as the realworld conditions are typically much more complex. As an alternative, we can represent this problem domain as a Bayesian network and create a network graph in BayesiaLab. In BayesiaLab, variables are represented as blue nodes and direct probabilistic relationships are shown as arcs. The direction of such arcs may
represent a causal assumption.
In our case, the network of the problem domain will look like this:
However, for now this only says, whether or not a crime has occurred will have a direct influence on the
probability of whether or not evidence is found.
To use this Bayesian network and BayesiaLab for inference, we also need to specify all known probabilities,
e.g. from crime statistics, from the expert witness, etc. We can enter these values via BayesiaLab’s Node Editor.
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This will associate a marginal distribution with Crime and a conditional probability distribution for Evidence as illustrated in the following.
In this format, BayesiaLab can carry out inference automatically. However, prior to observing any crime or
evidence, the prior probabilities would be shown by default in BayesiaLab’s Monitor Panel.
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In BayesiaLab, Monitors are small bar charts which display the distributions of any selected variable in the
network. For reference, the graphical user interface is shown in the screenshot below. The network and the
Monitors appear in the Graph Panel (left) and in the Monitor Panel (right) respectively.
Within BayesiaLab we can now simply carry out inference by observing evidence, i.e. by setting Evidence=“True”,
and BayesiaLab will automatically update the conditional probability distribution of Crime:
As we computed the probability of the cause given its effect, this represents a form of diagnosis.2 We have
now arrived at the same conclusion, except that BayesiaLab has performed all the necessary computations
for us.3
2
The term diagnosis is more common in the medical context, where a physician may determine the probability of a
specific illness, given certain symptoms. The direction of inference, from effect to cause, is the same though.
3
While the correctness of such probabilistic computations in BayesiaLab (and in other programs) are undisputed in the
scientific community, they are, like these puzzles, still viewed with skepticism by the general public. It is unfortunate
that it will presumably take many more years before these computations will find widespread acceptance and use in
court and in other areas of decision making.
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Paradoxes and Fallacies
Simpson’s Paradox
At the peak of the recent recession, Simpson’s Paradox made headlines again as the media inundated us
with countless statistics about the condition of the economy. However, some of the statistics seemed utterly
incongruent and thus undoubtedly generated conflicting interpretations, perhaps furthering policymakers’
already diverging views.
It becomes an even more immediate problem when Simpson’s Paradox rears its ugly head in the context of
medical studies, where it can suggest a false interpretation of a treatment effect.
We use an admittedly contrived example to illustrate this problem. A hypothetical type of cancer equally
effects men and women. A long-term study finds that a specific type of cancer therapy increases the remission rate from 40 to 50% among all treated patients (see table). Based on the study, this particular treatment is thus recommended for broader application.
Treatment
Yes
No
Remission
Yes
No
50%
50%
40%
60%
However, when examining patient records by gender, the remission rate for male patients — upon treatment — decreases from 70% to 60% and for female patients the remission rate declines from 30% to 20%
(see table). So, is this new therapy effective overall or not?
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Gender
Male
Female
Treatment
Yes
No
Yes
No
Remission
Yes
No
60%
40%
70%
30%
20%
80%
30%
70%
The answer lies in the fact that — in this example — there was an unequal application of the treatment to
men and women. More specifically, 75% of the male patients and only 25% of female patients received the
treatment. Although the reason for this imbalance is irrelevant for inference, one could imagine that side
effects of this treatment are much more severe for females, who thus seek alternatives therapies. As a result,
there is a greater share of men among the treated patients. Given that men also have a better recovery prospect with this type of cancer, the remission rate for the total patient population increases.
So, what is the true overall effect of this treatment? With a Bayesian network, the paradox can be easily
resolved, and the effect can be computed automatically. However, to create a Bayesian network for this
purpose, we first need to make specific assumptions regarding causality.4
With our knowledge of the domain, we can make such causal assumptions and thus define a causal network. As stated earlier, we assume that Gender has a causal effect on Remission (rather than Remission on
Gender), so we define the arc Gender ➝ Remission. We also assume that Treatment has a causal effect
(whether positive or negative) on Remission, which translates into Treatment ➝ Remission. Finally, we have
learned that Gender influences (causes) whether or not one would undergo Treatment, so we have Gender
➝ Treatment.
4
The concept of causality has been highly controversial over the last 100 years and for a long time it seemed entirely
banned from statistical literature. Causality has emerged from obscurity in recent decades in now plays a central role in
the study of Bayesian networks.
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Once we have this structure, we still need to enter all the marginal and conditional probabilities we have
observed. We can do so be specifying the values via BayesiaLab’s Node Editor. The following illustration
shows the network plus the tables associated with each node. For Gender, we have a one-dimensional table
(marginal probabilities only), for Treatment, a two-dimensional table (conditional probabilities, given Gender) and finally, for Remission, a three-dimensional table (conditional probabilities, given Gender and
Treatment).
Now the structure and the parameters of the Bayesian network are defined, and we can proceed to inference. The original statement about this domain was that, given Treatment and without specifying Gender,
total Remission increases from 40% to 50%. If this Bayesian network is a correct representation of our domain, it will need to return the proportions as we observed them originally.
By setting evidence on the Treatment node and not setting evidence to Gender, we can test this.
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In the bottom Monitors, we can now see that Remission indeed goes from 40% to 50%, but we also see
that, given Treatment, the proportion of men grows from 25% to 75% (top Monitors). This reflects the
omnidirectional inference property of Bayesian networks. Even though we were only looking for inference
on Remission, we inevitably saw another implication, namely, given Treatment, the balance of Gender also
changes.
When we now set evidence to Gender, e.g. Gender=“male”, we will confirm the seemingly paradoxical result, i.e. that Remission decreases.
For the sake of completeness, we repeat this for Gender=“female”:
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We have now verified that all the original statements are fully compatible with the network representation,
but the final answer remains elusive, i.e. how much does Treatment affect Remission in general? To answer
this we, must make a distinction between observational inference and causal inference. This is because of
the semantic difference of “given that we observe” versus “given that we do.” The former is strictly an observation, i.e. we focus on patients who received treatment, whereas the latter is an active intervention. The
answer to our question of the treatment effect then is inferring as to what would hypothetically happen,
“given that we do,” i.e. given that we force the treatment without permitting patients to self-select their
treatment. In the semantics of Bayesian networks, this means that there must not be a direct relationship
between Gender and Treatment. In other words, Treatment must not directly depend on Gender. In our
Bayesian network this can be done easily by mutilating the graph, i.e. deleting the arc connecting Gender
and Treatment or by fixing the distribution of Gender. BayesiaLab offers a very simple function to achieve
this, which is aptly named Intervention.
By intervening on the Treatment variable (and setting Treatment=“yes”), the causal Bayesian network is
modified as follows:
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Now we can observe what happens to Remission when we “do” Treatment, instead of just “observing”
Treatment.
In fact, Remission decreases from 50% to 40%, given that we “do” Treatment, and so we must conclude
that the new treatment is detrimental to the patients’ health.
Outside our made-up example, e.g. in real clinical trials, such “do” conditions are achieved with controlled,
randomized experiments, which allow investigators to determine the true effectiveness of new drugs. However, the real world is not a controlled experiment and self-selection is often inherent in many observational
studies. Without being conscious of Simpson’s Paradox, results can be easily perceived as the opposite of the
truth.
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The Monty Hall Problem
The Monty Hall Problem is a probability puzzle based on the American television game show Let’s Make a
Deal, originally hosted by Monty Hall.
In her book, The Power of Logical Thinking, Marylin vos Savant quotes cognitive psychologist Massimo Piattelli-Palmarini
as saying, “... no other statistical puzzle comes so close to fooling all the people all the time” and “that even Nobel physicists
systematically give the wrong answer, and that they insist on it,
and they are ready to berate in print those who propose the
right answer.”
Whereas some of the earlier descriptions of the game led to different interpretations of the problem, Krauss
and Wang (2003) state a fully unambiguous and mathematically explicit version of this problem:
“Suppose you’re on a game show, and you’re given the choice of three doors [and will win what is
behind the chosen door]. Behind one door is a car; behind the others, goats [i.e. silly prizes]. The
car and the goats were placed randomly behind the doors before the show. The rules of the game
show are as follows: After you have chosen a door, the door remains closed for the time being. The
game show host, Monty Hall, who knows what is behind the doors, now has to open one of the
two remaining doors, and the door he opens must have a goat behind it. If both remaining doors
have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door
with a goat, he will ask you to decide whether you want to stay with your first choice or to switch
to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a
goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to
change your choice?”
The vast majority of people intuitively believe that, when keeping their original choice, they will have a 50/
50 chance of winning, which turns out to be incorrect. There are a number of explanations, which resolve
this paradox and a several of them are detailed on Wikipedia as well as on Marylin vos Savant’s website.
Rather than reiterating those explanations, we will demonstrate how a Bayesian network can quickly and
simply produce the correct answer.
The game description from above can be fully expressed with the structure and the parameters of a Bayesian
network. In terms of structure, we know that only one variable is dependent on other variables, and that is
Monty’s choice as to which door to open. His choice is a function of (or caused by) the contestant’s first
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choice and the actual location of the valuable prize, which Monty knows. Other than that, there are no direct dependencies. As such, we can introduce two causal arcs pointing to Monty’s Choice.5
For the parameters of this network, we simply need to restate the rules of the game as marginal and conditional probabilities and enter them via BayesiaLab’s Node Editor. As the prize is randomly placed, each of
the three doors is equally likely to be the prize-winning door, thus assigning a one-third probability to each
door. Without knowing the preferences of the contestant, we assume that there will also be a one-third
chance of him or her picking a specific door (the a-priori probability of the contestant’s choice actually does
not matter).
The game rules determining Monty’s Choice can be expressed in table form and is entered that way in the
Node Editor:
The fully specified network is shown below with its associated marginal and conditional probability tables.
5
In this case there can be no doubt about the direction of the arcs, as both Contestant’s Choice and Winning Door are
determined before Monty’s Choice is set. A causal arc going back in time is obviously not possible.
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Let’s go through this example, and for the sake of argument assume that the contestant picks Door 1. If the
prize is actually behind Door 1, Monty will pick either one of the other doors at random, so Door 2 and
Door 3 will both have a 50% probability of being opened. If the prize, however, is behind Door 2, Monty
will certainly not open Door 2, but rather open Door 3, implying a 100% probability for the latter. Finally,
if the prize is behind Door 3, Monty must open Door 2. Examples for other initial door choices of the contestant follow analogously. The entire logic is fully expressed in the conditional probability table for the
node Monty’s Choice.
With all the parameters specified, we now have a Bayesian network representing our problem domain, and
this provides us with our desired inference tool. We will now observe the inference progression from the
contestant’s perspective as the game evolves.
Before the game starts, all probabilities are uniformly distributed. At this point, our inference tool is of no
help and the contestant’s random pick of a door is as good as any other choice.
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So, let’s assume the contestant picks Door 1, which in Bayesian network terms means setting the node Contestant’s Choice to state 1. From the rules (and the conditional probability table) we know that Monty never
opens the door the contestant picked, so the probability of opening Door 1 is zero. As the contestant has no
knowledge about the true location of the prize, he only knows that one of the other doors will be opened
and that, at this particular point in time, his belief should be 50/50 regarding either door. This is precisely
what we can see in the Monitor Panel, once we set Contestant’s Choice to state 1
Now, given his knowledge about the location of the prize, Monty responds and picks Door 2, which will
inevitably reveal a worthless prize.
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The crucial question is, how should this observation change our contestant’s belief in the probabilities of the
prize being behind either Door 1 or Door 3? Should the contestant update his beliefs about the location of
the door, given his first choice plus Monty’s subsequent choice?
BayesiaLab can compute this new probability distribution by setting evidence to the node Monty’s Choice,
i.e. Monty’s Choice=2, which we have just observed.
Given Monty’s choice of Door 2, BayesiaLab computes a two-thirds probability of the prize being behind
Door 3 and only a one-third probability of being behind Door 1. So, the contestant’s rational choice would
be to change his choice of doors, and not stick to his original pick.
How does BayesiaLab determine this? BayesiaLab consequently applies Bayes’ Rule to compute the new
posterior probabilities, given the emerging evidence. Without going into further detail, the key point is that
setting evidence to Monty’s Choice renders Contestant’s Choice and Winning Door dependent and allows
information to “flow” across nodes and update Winning Door.
Readers, who are in still doubt, may want to experiment and repeatedly play this game, perhaps with three
cups and a coin. After several rounds, one will find that the probability of winning converges to a value of
2/3 if the recommended switching policy is applied consistently. If not, the chance of winning remains at 1/
3.
Conclusion
For reasons we have not discussed here, the cognitive skills of humans are inherently limited when it comes
to dealing with numerous pieces of evidence, especially when such pieces of evidence represent uncertain
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observations. Bayesian networks are a very practical tool for carrying out inference in these situations. With
programs like BayesiaLab, rephrasing the paradoxical problem domain into a causal model is a relatively
easy task, consisting of individually intuitive steps. Given such a model, carrying out inference, which is so
tricky and counterintuitive for humans, becomes entirely automatic.
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Appendix
Bayes’ Theorem
Bayes’ theorem relates the conditional and marginal probabilities of discrete events A and B, provided that
the probability of B does not equal zero:
P(A∣B) =
P(B∣A)P(A)
P(B)
In Bayes’ theorem, each probability has a conventional name:
P(A) is the prior probability (or “unconditional” or “marginal” probability) of A. It is “prior” in the sense
that it does not take into account any information about B. The unconditional probability P(A) was called
“a priori” by Ronald A. Fisher.
P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
P(B|A) is the conditional probability of B given A. It is also called the likelihood.
P(B) is the prior or marginal probability of B.
Bayes theorem in this form gives a mathematical representation of how the conditional probability of event
A given B is related to the converse conditional probability of B given A.
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References
Fountain, J., and P. Gunby. “Ambiguity, the Certainty Illusion, and Gigerenzer’s Natural Frequency Approach to Reasoning with Inverse Probabilities” (2010).
Glymour, Clark. The Mind’s Arrows: Bayes Nets and Graphical Causal Models in Psychology. The MIT
Press, 2001.
Kahneman, Daniel, Paul Slovic, and Amos Tversky. Judgment under Uncertainty: Heuristics and Biases. 1st
ed. Cambridge University Press, 1982.
Krauss, S., and X. T. Wang. “The Psychology of the Monty Hall Problem: Discovering Psychological
Mechanisms for Solving a Tenacious Brain Teaser* 1,* 2.” Journal of Experimental Psychology: General 132, no. 1 (2003): 3–22.
“Monty Hall Problem.” http://en.wikipedia.org/wiki/Monty_Hall_problem.
Nobles, R., and D. Schiff. “Misleading statistics within criminal trials.” Significance 2, no. 1 (2005): 17–19.
Pearl, Judea. Causality: Models, Reasoning and Inference. 2nd ed. Cambridge University Press, 2009.
vos Savant, Marilyn. “Game Show Problem.” http://www.marilynvossavant.com/articles/gameshow.html.
———. The Power of Logical Thinking: Easy Lessons in the Art of Reasoning...and Hard Facts About Its
Absence in Our Lives. St. Martin’s Griffin, 1997.
Tuna, Cari. “When Combined Data Reveal the Flaw of Averages.” wsj.com, December 2, 2009, sec. The
Numbers Guy. http://online.wsj.com/article/SB125970744553071829.html#articleTabs%3Darticle.
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Bayesia Singapore Pte. Ltd.
20 Cecil Street
#14-01, Equity Plaza
Singapore 049705
Phone: +65 3158 2690
[email protected]
www.bayesia.sg
Bayesia S.A.S.
6, rue Léonard de Vinci
BP 119
53001 Laval Cedex
France
Phone: +33(0)2 43 49 75 69
[email protected]
www.bayesia.com
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