The recent tragedy at Fort Hood has people griping about missed signals that could have been used to prevent the attack.  However, I will argue that is likely to be impossible to ever have a system that can screen out all terrorists without also flagging a lot of innocent people.  The calculation is a simple exercise in probability theory that is often given in first year statistic classes.

Suppose we have a system in place that gives a yes Y or no response of whether or not a person is a terrorist T.  Let P(T) be the probablity that a given person is a terrorist,  P(T|Y) be the probability that a person  is a terrorist given that the test said yes.  Thus P(~T|Y)=1-P(T|Y) is the probability that one is not a terrorist even though the test said so.  Using Bayes theorem we have that

P(~T|Y)=P(Y|~T) P(~T)/P(Y)  (*)

where P(Y)=P(Y|T)P(T) + P(Y|~T)P(~T) is the probability of getting a yes result.   Now, the probability of being a terrorist is very low.   Out of the 300 million or so people in the US a small number are probably potential terrorists.  The US military has over a million people on active service.   Hence, the probability of not being a terrorist is very high.

From (*),  we see that in order to have a low probability of flagging an innocent person we need to have  P(Y|~T)P(~T)<< P(Y|T)P(T), or P(Y|~T)<< P(Y|T) P(T)/P(~T).  Since  P(T) is very small, P(T)/P(~T)~ P(T),   so if the true positive probability P(Y|T) was near one (i.e. a test that catches all terrorists), we need the false positive probability P(Y|~T) to be much smaller than the probability of having a terrorist, which means we need a test that gives false positives at a rate of less than 1 in a million.  The problem is that the true positive and false positive probabilities will be correlated.  The more sensitive the test the more likely it is to get a false positive.  So if you set your threshold to be very low so P(Y|T) is very high (i.e. make sure you never miss a terrorist), you’ll most certainly have P(Y|~T) to also be high.  I doubt you’ll ever have a test where P(Y|T) is near one while P(Y|~T) is less than one in a million.   So basically, if we want to catch all the terrorists, we’ll also have to flag a lot of innocent people.

Not too long ago, Proctor & Gamble marketed Head and Shoulders shampoo with twenty-six variations. Sales were good, but not maximal. To increase sales of their shampoo, they decided to reduce the number of variations from twenty-six to fifteen. Result: Sales increased 10% [1].

But does that make sense? Popular notion would seem that the more choices an individual can make, the happier they will be. Don’t you like to have the option to choose between a small, medium, large, or extra-large drink? It turns out that the popular notion isn’t wrong, sometimes it’s just applied at the wrong times.

When given the option to choose, increasing the number of options from 2 to 6 increases the satisfaction of the individual and reduces the amount of time to make said decision [1]. But at what point does the addition of options have diminishing returns? George Miller, a cognitive psychologist at Princeton in 1956, found that the optimal number of choices are 7, plus or minus 2 [2].

This 7 ± 2 rule has been widely cited, and it may be the reason that many fast food restaurants have pared down the number of value meals that are offered to fit within this range. When the number of options is higher than 9, individuals will feel overwhelmed and burdened by the worry that comes with making a “wrong” choice.

Residents of the United States are faced with so many decisions on a daily basis, including decisions that are likely to affect loved ones, that the decision-making process becomes demotivating [3]. Up to this point (2000), there has been research that focused on the 7 ± 2 rule, options ranging from 2 to 6, but no research on options ranging in count from 6 to 24 or 6 to 30.

A study by Sheena Iyengar of Columbia University and Mark Lepper of Stanford University specifically looks at the effects of choice and has interesting findings. Iyengar and Lepper performed three studies observing participants initial reaction, satisfaction, and decision-making when presented with extensive and limited selections.

One of their studies focused on the sale of exotic jams. They stationed a sample booth in a gourmet grocery store and offered varying numbers of jam to sample. If a customer approached the sample booth, they received a coupon towards a jam, and purchases of the jam were recorded to determine the effect of the sampling booth.

Table 1

Those results show the following percentages:

Table 2

These results are very interesting. Although the number of customers that approached the extensive selection (60%) was much higher than the limited selection (40%), these customers did not purchase the jam. In fact, the customers that approached the limited selection were 10 times more likely to purchase.

Given those percentages, can we turn the data on its head and determine the effect that the number of flavors presented had, given that the customer was going to purchase the jam?

To answer the question, we can use the naïve Bayes theorem [4]. The naïve Bayes theorem provides a way to find the posterior probability given a pre- and post-condition. We can use the percentages given in Table 2 and apply them to the theorem:

The probabilities for purchasing without sampling were  not given, and thus we will use a weighted average of the purchasers to estimate the probability at 0.095 ((4/386) + (31/368)). These numbers are chosen because we will have to assume that regardless of the number of flavors present at the sample booth, these customers were already going to make a purchase. Ideally, we would have in our data set the number of customers who purchased regardless of stopping at the sample booth.

P₆(sample|purchase) = (P₆(purchase|sample) * P₆(sample))/(P₆(purchase)) = (0.3 * 0.4)/0.095 = 1.26
P₂₄(sample|purchase) = (P₂₄(purchase|sample) * P₂₄(sample))/(P₂₄(purchase)) = (0.03 * 0.6)/0.095 = 0.19

Using the naïve Bayes Theorem, we are able to find that when customers were buying jam, they were 663% more likely to stop and sample jam if the sample booth only contained 6 flavors compared to 24 flavors. The increase in choice may have at first glance brought many more customers to the sample booth, yet the customers that approached the booth when 24 flavors were presented may never have intended on purchasing the jam.

So Miller’s rule says that 7 ± 2 is the right number of choices, and Iyengar and Lepper make a case that happiness does not increase with choices. Right now, we are encountering too many choices. Most of us would probably choose the same options, so we might as well be happier about our choices. Therefore, if our goal is to make the customer happier, the selection should be limited before they encounter their decision. In the end, we will all be happier.

[1] http://www.columbia.edu/~ss957/articles/Choice_is_Demotivating.pdf
[2] http://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two
[3] http://www.ucpress.edu/books/pages/5572001.php
[4] http://en.wikipedia.org/wiki/Naive_Bayes_classifier

Il secondo giorno di meeting era il 18 ottobre. Lo so, le date sono sfasate ma:

1. sto scrivendo in ritardo;
2. l’ora dei post è rimasta quella italiana, mentre qui sono 7 ore indietro.

Altra giornata piena. Bellissimo il seminario di Daniel Wolpert: “Moving in an Uncertain World: Computational Principles of Human Motor Control“.

Esposizione chiara e divertente dell’integrazione sensorimotoria. Approccio bayesiano. Mostrati anni di esperimenti.

Tra le tante cose alcuni punti:

• Secondo Wolpert capire il movimento significa capire il cervello. Infatti sostiene che l’unico motivo che abbiamo di avere un sistema nervoso è quello di muoverci. A parte sudare.
• La distribuzione di probabilità dei movimenti varia per diversi movimenti. [Una premessa: anche per l'esecuzione dei comandi motori, come per la percezione, c'è incertezza e rumore]. Posso effettuare movimenti diversi per arrivare ad uno stesso scopo. Avrò meno incertezza sul raggiungimento dello scopo se scelgo il movimento con la distribuzione di probabilità più ristretta.
• Le informazioni sensoriali continuano ad essere processate anche quando non c’è più lo stimolo che le ha causate. Questo permette di migliorare una scelta, ma in ritardo, e quindi a cambiarla anche dopo che l’azione è iniziata.
• Se il sistema usa una funzione di costo/beneficio differente prenderà anche decisioni diverse.

Su quest’ultimo punto: mi chiedo se sia possibile utilizzarlo anche a livello comportamentale per migliorare gli allenamenti o la performance. Siccome il sistema cerca di ottimizzare il suo comportamento per minimizzare i costi dell’effetto della sua azione, se nell’allenamento do peso di più o penalizzo maggiormente certi certi aspetti piuttosto che altri cambio ciò come il sistema cerca cambiare per ottimizzare il risultato. Detto così è un po’ fumoso, a livello intuitivo ho chiaro il senso, ma devo elaborarlo di più perché possa scriverne esplicitamente.

Un scelta della bibliografia citata nel seminario:

• Weiskrantz L, Elliott J, Darlington C. Preliminary observations on tickling oneself. Nature 1971, 230:598-599.
• Shergill SS, Bays PM, Frith CD, Wolpert DM. Two eyes for an eye: the neuroscience of force escalation. Science 2003, 301:187
• Shergill SS, Samson G, Bays PM, Frith CD, Wolpert DM. Evidence for sensory prediction deficits in schizophrenia. Am J Psychiatry 2005, 162:2384-6.

La bibliografia continua, andrò avanti. Ora ho problemi tecnici a recuperarla.

I’m currently at the Mathematical Biosciences Institute for a workshop on Computational challenges in integrative biological modeling.  The slides for my talk on using Bayesian methods for parameter estimation and model comparison are here.

Abstract: Differential equation models are often used to model biological systems. An important and difficult problem is how to estimate parameters and decide which model among possible models is the best. I will argue that Bayesian inference provides a self-consistent framework to do both tasks. In particular, Bayesian parameter estimation provides a natural measure of parameter sensitivity and Bayesian model comparison automatically evaluates models by rewarding fit to the data while penalizing the number of parameters. I will give examples of employing these approaches on ODE and PDE models.

KODE føler seg lurt til å få panikk av en svineforkjølelse-artikkel i New York Times. Jeg kjenner meg igjen.

Det er noe som heter at “ekstraordinære påstander krever ekstraordinære bevis”, at man ikke skal tro alt man hører, at man alltid må bruke sunt bondevett, at man ikke skal høre på ekstremister, osv.

Dette finnes også i statistikk, og kalles, på engelsk, Bayesian average (igjen takk til Wikipedia):

A Bayesian average is a method of estimating the mean of a population consistent with Bayesian interpretation, where instead of estimating the mean strictly from the available data set, other existing information related to that data set may also be incorporated into the calculation in order to minimize the impact of large deviations within the data set, or to assert a default value when the data set is small.

Det høres sikkert komplisert ut, men det blir lettere å forstå med dette eksempelet:

For example, in a calculation of an average review score of a book where only two reviews are available, both giving scores of 10, a normal average score would be 10. However, as only two reviews are available, 10 may not represent the true average had more reviews been available. The review site may instead calculate a Bayesian average of this score by adding the average review score of all books in the store to the calculation. For example, by adding five scores of 7 each, the Bayesian average becomes 7.86 instead of 10, which the review site would hope that it will better represent the quality of the book.

Det blir subjektivt hvor mye man ønsker å trekke ned gjennomsnittet på denne måten. Det kommer her bl.a. an på hvor objektive/dyktige, og representative, man mener anmelderne er (altså, vekt og relevans).

I tilfellet svineinfluensa har KODE konsultert kun én kilde, New York Times, som gir swineflu 10/10 poeng på panikk-skalaen. KODE skulle kanskje ha forstått hvor lite sannsynlig det var at dette var representativt og at han dermed burde få et panikkanfall tilsvarende 10/10 poeng, både ut fra kildens seriøsitet, hva man har kunnet lese andre steder, og ut fra prinsippet at det er særdeles få fenomener som skal gi 10/10 panikkpoeng.

Problemer med å vurdere og gjennomskue falske ekstraordinære påstander dukker gjerne opp når man stoler på veldig få kilder og dermed får et lite datagrunnlag (“få anmeldere”). I de aller fleste saker bør man faktisk stole på at verden henger på greip og at vanlige folk – eller på mer spesialiserte områder i det minste eksperter – har noenlunde snøring.

It is amazing how a very simple mathematical formula can hold such a world of detail and be such a powerful tool in understanding reality.

One such formula is Bayes’ Theorem

This theorem tells us about the conditional probability of one event given that another event has been observed, what that is really about is the following:

How do we assess the likelihood of an event given a piece of evidence that may or may not have bearing on it?

The idea can be built up quite simply. let’s think of two events called A and B. The probabilities of each of these happening are represented as P(A) and P(B).

We’ll immediately make this concrete to explain it best. Let’s denote as follows

A : John is guilty of the murder
B : John was found at the scene after the dead body was found with blood on his hands

A has yet to be established, but we know B has already happened.

Well Bayes’s Theorem gives us one possible way to do some calculation about this. Time for notation…

P(A|B) is how we write “The probability of A happening given that B has happened.

it turns out that P(A|B) can be expressed as

P(A|B) = P(A) * ( P(B|A) / P(B) )

So in this case P(A) will be the probability of a freely chosen individual in the sample being guilty of a murder. This is the basic rate when we have absolutely no evidence to narrow it down. In other words P(A) will be the murder rate in whatever country we are in.

P(B) will be the likelihood of a random individual being found with blood on their hands at a crime scene

and P(B|A) is the probability of someone being found with blood on their hands when it is given and therefore certain they have just committed a murder. In other words this is a quantity that is the correlation or degree of connection between murder and having blood on your hands.

To look at our example again, maybe if someone worked at an abbatoir or butcher’s they might frequently be found with blood on their hands, so it is not self evident that the given evidence guarantees the probability P(A) must be 1. However it does seem likely that this kind of evidence is compelling, therefore we expect the correlation of bloody hands with murders to be quite high. P(B) is the denominator in the right hand term though, so if B is rare and therefore P(B) small then that enhances our evidence’s power to convict. P(A) is the starting likelihood of john being a murderer before we found the evidence. So you can see that

P(B|A) / P(B) is the amount by a factor of which our evidence alters the likelihood of A. If P(B|A) is small then our evidence may not help us that much, unless P(B) is even smaller. If P(B|A) is big, i.e. if the two events of bloody hands and guilt are strongly correlated then the evidence may be helpful. If there’s slight correlation but B is a common event then once again the evidence may not be much help. You can play around with the figures yourself but its such a beautifully simple formula that it becomes quite easy to visualise.

One of the noted applications of Bayesian Reasoning of this kind is in using healthcare data about the error rates of clinical tests. A scientist online quoted an example of the Mammogram as a test for breast cancer, and to what extent we can expect a positive mammogram test to lead us to believe that the subject has breast cancer. Interestingly the figures work out to show that a small but appreciable false positive rate and a similarly small nonzero false negative rate seriously diminish the validity of the test. Here’s how it goes:

1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

quoted from http://yudkowsky.net/rational/bayes

well if A is having cancer and B is testing positive then P(B|A) should be the frequency of people who test positive while definitely and verifiably having cancer.

so P(A), the incidence of cancer in the group, is clearly 0.01

P(B) is a little subtler, since it has to include the false positives rate, I make

P(B) = (0.01 * 0.8) + (0.096 * (1- 0.01)) = 0.10304

this is the incidence of positive tests, real illness notwithstanding

and finally

P(B|A) = 0.8

which is 1 minus the false negative rate (number of people with cancer who don’t test positive)

so here:

P(A|B) = 0.01 * (0.8 / 0.10304) = 0.07763 = 7.763%

In other words the test is really not a conclusively strong piece of evidence. The catch is that there are so many more women who don’t have breast cancer that even a relatively small proportion of false positives (people who the test says have cancer but who don’t in reality) changes the figures quite drastically.

So there we have it ! The great thing about this stuff is that there is such a myriad profusion of examples we can think up. Any area where some basic stats are known can really elucidate and sharpen our understanding of how evidence based reasoning works out in practice.

I have also personally found that going to the trouble of some rule of thumb calculations can even help allay personal fears about nasty and dangerous events, which duly do prove to be rather unlikely. Maths beats paranoia every time !!

This past Sunday, economist Paul Krugman was lamenting in a book review of Justin Fox’s book “Myth of the Rational Market” (which he liked very much) that despite this current financial crisis and previous crises, like the failure of the hedge fund Long Term Capital Management, people still believe in efficient markets as strongly as ever. The efficient market hypothesis is the basis of most of modern finance and assumes that the price of a security is always correct and that you can never beat the market.  So artificial bubbles should never occur.  Krugman wonders what it will take to ever change people’s minds.

I want to show here that there might be no amount of evidence that will ever change their minds and they can still be perfectly rational in the Bayesian sense.  The argument can also apply to all other controversial topics.  I think it is generally believed in intellectual circles that the reason there is so much disagreement on these issues is that the other side is either stupid, deluded or irrational.   I want to point out that believing in something completely wrong even in the face of overwhelming evidence may arise in perfectly rational beings.  That is not to say that faulty reasoning does not exist and can be dangerous.  It just explains why two perfectly reasonable and intelligent people can disagree so alarmingly.

Consider the very simple case of a hypothesis H, like “the market is not efficient” and there is some data D, like a financial crisis.  Then from Bayes rule, the probability that the hypothesis is true given the data is , where P(D|H) is the likelihood function (probability of obtaining the data given the hypothesis is true), P(H) is the prior probability the hypothesis is true, and P(D) is the probability of obtaining the data.  Thus the odds that the hypothesis is true to it being false is

So, you see that even if two people have identical likelihood functions (i.e. reasoning ability) and have the same data, they can still come to completely different conclusions depending on their priors.  For example, let’s say two people agree that the odds of a crisis occurring given that efficient markets are false is 100 to 1.  So whatever the prior odds against an efficient market was before, it is now 100 times greater after the crisis.  So for someone who may have believed that efficient markets had even odds of being false now believes it is 100 to 1 that it wrong.  But, someone who originally believed that the odds of an efficient market were false  was one in a million now believes it is one in ten thousand.  Hence, given enough events they will eventually change their minds.  However, suppose there is a person who believed that the probability of an efficient market is false is zero.  Then they are completely unaffected by the data and no amount of data can ever convince them.  If a hypothesis has zero prior support then it can never be validated no matter what the data.

You could argue that a zero prior is faulty to start with but that is not a failure of reasoning.  In fact, it is easy to see how a prior could be zero.  Suppose you are a naive student and you take a class from a Fama or a Miller who implicitly assumes that the market is efficient. You could easily end up believing that it is a law of nature like gravity.  There could even be an amplification effect in that the professor may have some doubt about the idea but for pedagogical or other purposes will not bring it up in her class.   Then the next generation is even more certain and eventually it becomes dogma.

So why are there efficient market doubters?  Well I think that there are probably some neural mechanisms that sets a minimum doubt level in every person.  Some people have complete certainty in their beliefs while others have doubts about everything.  I believe that this doubt level is innate and could be related to genes governing certain ion channels in the brain.  So some of the students will not completely believe in the efficient market.   However, given that doubt level seems to be broadly distributed in the population there must be advantages for maintaining diversity in a population.  A community of pure believers is dangerous (e.g. Jonestown) but one of doubters may end up starving to death because they can never decide on what to do.  A balance of the two may be necessary to get things done but also have a reality check.  This also means that some wrong ideas will only disappear when the zero doubt holders take them to the grave.

Approaches to AI

The introductory chapter starts with a four-fold categorization of the approaches that have been taken in AI:

1. Thinking like Humans
Cognitive science pretty much sums up this category. The idea is to understand how humans make intelligent decisions (using neuroscience, psychology, etc) and model it on an artificial system.

2. Acting like Humans
Here the focus is on the results produced by humans. Any approach which has as it’s aim passing the Turing test can be considered in this category. AI approaches that can be seen as attempts to completely build an ‘artificial human’ are considered to be in this group: natural language processing, knowledge representation, automated reasoning, machine learning, vision, robotics, etc.

3. Thinking Rationally
The logicist tradition occupies this category. Focus: represent knowledge, make logical inferences on it to make decisions.

4. Acting Rationally
The main approach considered in the book. Any approach that attempts to build an agent whose actions can be considered rational can be considered to be in this category.

For the first two categories of approaches, success would be measured with respect to a human – which is quite a difficult task. With the latter two approaches, it would be measured to a more well defined rationality.

In my opinion, it is important to notice that the categorization is not absolute and there is considerable overlap; particularly, the last two categories do not have a clear boundary. From one perspective, to “think rationally” is to decide how to infer the most rational action to take, and hence, category 3 is included in category 4. On the other hand, if it can be considered that any agent that makes a rational action has implicitly made a rational decision, then category 4 is included in category 3.

The book cites the following example: “For example, recoiling from a hot stove is a reflex action that is usually more successful than a slower action taken after careful deliberation”. I disagree with this example, since the decision to recoil is a rational decision, taken after inferring that there is not enough time to engage in more complex decision making and that an action must be taken as quickly as possible. i.e. The example is a case of making logical inferences with temporal logic.

Question: To which category do neural network approaches fall?

Philosophy

Since Aristotle’s syllogisms there have been attempts to create formal reasoning systems. Philosophy discusses the limits and possiblities of logic, and also of AI.

Some important terms introduced are:
1. Dualism: the idea that there is a immaterial part to the universe, that the mind is not a manifestation of the physical brain. (Was this Descartes’ big mistake?)

2. Materialism: the physical universe is all that exists.

3. Empiricism: knowledge is distilled from one’s experiences.

4. Hume’s principle of induction: general rules are aquuired by exposure to repeated association between their elements. (Is this the same as pattern recognition?)

5. Logical positivism: all knowledge can be characterised by logical theories, and all meaningful statements of that logic system can be verified or falsified. Developed by the infamous Vienna Circle.

6. Confirmation Theory: attempts to understand how to do induction; i.e. how knowledge can be aquired from experience.

One good example of how Philosphy has lended to concrete AI systems is Newell and Simons’ General Problem Solver (GPS). The basic regression planning algorithm used in GPS is none other than the one proposed by Aristotle: consider what the outcome is, and plan backwards from it – see what actions lead to the outcome, and then what actions lead to those actions, and so on.

Goal based analysis is discussed in Philosophy with respect to the question of understanding the relation between thinking and acting.

Mathematics

The contribution from mathematics can roughly be categorised into three areas:

1. Logic
While philosophy gave birth to logic, the mathematical development of logic into a formal system was what enabled it to become a strong tool that can be used for AI.

2. Computation
Decidability
This is linked to logic and number theory. Of particular interest is Hilbert’s decision problem, which questions whether it was possible to find an algorithm to decide the truth value of any logical proposition involving the natural numbers – i.e. whether there were limits to the power of effective proof procedures.

Computability
Gödel’s findings tell that
1) any statement expressed in first order logic that is true can be proved of it’s truth
2) first order logic is not powerful enough to capture the principle of mathematical induction needed to characterise the natural numbers
3) incompleteness theorem: any language expressive enough to capture the natural numbers fully (i.e. describe all the properties of the natural numbers) has, in that language, statements that are true, but their truth cannot be established by an algorithm. This means that some functions on the natural numbers cannot be computed by an algorithm.

Turing brought the idea of computability – that is, to find which functions can be computed and which cannot. The Church-Turing thesis presents the idea of a Turing machine that is capable of computing any computable function.

Tractability
From a practical point of view, tractability is a much more useful concept to study, because even if a function is computable theoretically, it can be practically uncomputable because the resources required increase exponentially with respect to input size. Such problems are said to be intractable.

Although it has not been proven (it is one of the remaining Millenium problems) it is generally assumed that NP-complete problems are intractable.

3. Probability
The most important contribution from probability is Bayesian analysis, borne out of Baye’s theorem which shows how probabilities change when new conditions are added to an event. Bayesian analysis forms the basis of most approaches to dealing with uncertainity in AI.

Economics

1. Decision Theory
Utility theory investiages how decisions can be made leading to a peferred outcome; utility theory combines with probability results in decision theory, where decisions made under uncertain conditions can be investigated. Here probability captures the decision maker’s environment, and it is assumed that the decision maker’s world is not affected by others.

2. Game Theory
If the decision maker also needs to consider what the other decision makers are doing, then such problems are studied in game theory.

3. Operations Research
One important contribution from OR to AI is Markovian decision processes, where a number of sequencially made decisions are studied.

Note that decision theory, game theory and operations research are fields that are difficult to classify only as either mathematics or economics, as they are a hybrid of both.

Neuroscience

Neuroscience can help in discovering how our brains manage to function as intelligent agents. Particularly the invention of fMRI has been very helpful in monitoring brain activity.

Psychology

1. Behaviourism
Behaviourism rejects mental introspection and only considers objective results of a psychological experiment. While behaviourism has helped to understand animal (non-human) behaviour, it has been less successful at understanding human behaviour.

When you look at it, this should be expected from behaviourism. Behaviourism treats the intelligent-process of the agent as a black box and looks at only the inputs (percepts from the environment) and outputs (actions of the animal). It is reasonable to assume that the function which takes place inside the black box is simple and animals and much more complex in humans. Thus it would be easy the guess the function for animals, but not for humans.

2. Cognitive Psychology
Cognitive psychology, in contrast, attempts to understand exactly the functionality that is taking place inside the brain, and does not exclude mental introspection (e.g. examining a persons beliefs and goals). Such mental introspection can be considered to be parts of a virtual representation of the world created in the brain.

Cognitive science, where cognitive psychological models are developed as computed models, thus pay a lot of attention to how the external world can be represented in an intelligent agent.

Control Theory and Cybernetics

Control theory explores how an automated system can monitor and regulate itself. Cybernetics, from an AI perspective is mainly the application of control theory to computational models of cognition to produce AI.

Due to the different areas of mathematics used in control theory and AI, there is some gap between the two fields. Control theory is built using calculus and matrix algebra, whereas AI (at least traditionally) used logic and computation. The problems of language, vision and planning, that were considered from AI perspectives, fell outside the domain of control theory due to the different mathematical tools used.

Linguistics

Chomsky introduced the idea of syntactic structures, which could capture the potential of a language so that it could be modeled computationally. This introduced linguistics to AI, resulting in the field of natural language processing. The main obstacle in NLP is understanding the context and subjec matter of the language (which is required due to the ambigious nature of human language), and thus knowledge representation is the field devoted to studying how information about the world can be captured into a computational structure so the information can be utilized by an intelligent agent.

Dear Colleagues,

The QLD Branch in conjunction with QUT and the National Office of the Statistical Society of Australia is running a a Three-day course on Practical Bayes for Beginners, presented by Professor Kerrie Mengersen of the Centre for Data Analysis, Modelling & Computation, QUT.

This will take place at the University of Queensland, Room S429, Hartley Teakle Building #83 University of Queensland, St Lucia, Brisbane on the 26th – 28th of August, 2009.

Given the trying times of the last year, as a branch and society, we really want to make this a success so I’m asking you to consider helping in the following ways:

1) Attending the workshop.
2) If not attending passing the link think people you know who might be interested in attending.
3) Promoting the workshop within your organisation.
4) If you familiar with WinBugs and R and can assist as a ‘technical tutor’ during the workshop let me know. You will be given a free place on the workshop.

It might be worth passing on to any interested parties, that by attending the workshop as a non-member of the SSAI, you are given membership of the society as part of your registration fee.

More details are available on the SSAI webiste: https://www.statsoc.org.au/CPD2

Kind Regards,

Helen

Helen Johnson, PhD | Biostatistician | Research Methods Group | Institute of Health & Biomedical Innovation | Queensland University of Technology | 60 Musk Ave, Kelvin Grove, Queensland, 4059, Australia | t: +61 7 3138 6053 | f: +61 7 3138 6030 | e h.johnson@qut.edu.au | w: www.ihbi.qut.edu.au

From 2007 International Society for the History, Philosophy, and Social Science of Biology in Exeter, England.

Wikipedia calls this a classic paradox of elementary probability theory.

Who was this Bertrand?

Taken from Wikipedia.org:

Card version

Suppose you have three cards:

• a black card that is black on both sides,
• a white card that is white on both sides, and
• a mixed card that is black on one side and white on the other.

You put all of the cards in a hat, pull one out at random, and place it on a table. The side facing up is black. What are the odds that the other side is also black?

The answer is that the other side is black with probability 2/3. However, common intuition suggests a probability of 1/2 because across all the cards, there are 3 white, 3 black. However, many people forget to eliminate the possibility of the “white card” in this situation (i.e. the card they flipped CANNOT be the “white card” because a black side was turned over).

In a survey of 53 Psychology freshmen taking an introductory probability course, 35 incorrectly responded 1/2; only 3 students correctly responded 2/3.^[1]

Preliminaries

To solve the problem, either formally or informally, we must assign probabilities to the events of drawing each of the six faces of the three cards. These probabilities could conceivably be very different; perhaps the white card is larger than the black card, or the black side of the mixed card is heavier than the white side. The statement of the question does not explicitly address these concerns. The only constraints implied by the Kolmogorov axioms are that the probabilities are all non-negative, and they sum to 1.

The custom in problems when one literally pulls objects from a hat is to assume that all the drawing probabilities are equal. This forces the probability of drawing each side to be 1/6, and so the probability of drawing a given card is 1/3. In particular, the probability of drawing the double-white card is 1/3, and the probability of drawing a different card is 2/3.

In our question, however, you have already selected a card from the hat and it shows a black face. At first glance it appears that there is a 50/50 chance (ie. probability 1/2) that the other side of the card is black, since there are two cards it might be: the black and the mixed. However, this reasoning fails to exploit all of your information; you know not only that the card on the table has a black face, but also that one of its black faces is facing you.

Solutions:

Intuition

Intuition tells you that you are choosing a card at random. However, you are actually choosing a face at random. There are 6 faces, of which 3 faces are white and 3 faces are black. Two of the 3 black faces belong to the same card. The chance of choosing one of those 2 faces is 2/3. Therefore, the chance of flipping the card over and finding another black face is also 2/3. Another way of thinking about it is that the problem is not about the chance that the other side is black, it’s about the chance that you drew the all black card. If you drew a black face, then it’s twice as likely that that face belongs to the black card than the mixed card.

Labels

One solution method is to label the card faces, for example numbers 1 through 6.^[2] Label the faces of the black card 1 and 2; label the faces of the mixed card 3 (black) and 4 (white); and label the faces of the white card 5 and 6. The observed black face could be 1, 2, or 3, all equally likely; if it is 1 or 2, the other side is black, and if it is 3, the other side is white. The probability that the other side is black is 2/3.

Bayes’ theorem

Given that the shown face is black, the other face is black if and only if the card is the black card. If the black card is drawn, a black face is shown with probability 1. The total probability of seeing a black face is 1/2; the total probability of drawing the black card is 1/3. By Bayes’ theorem,^[3] the conditional probability of having drawn the black card, given that a black face is showing, is

Eliminating the white card

Although the incorrect solution reasons that the white card is eliminated, one can also use that information in a correct solution. Modifying the previous method, given that the white card is not drawn, the probability of seeing a black face is 3/4, and the probability of drawing the black card is 1/2. The conditional probability of having drawn the black card, given that a black face is showing, is

Symmetry

The probability (without considering the individual colors) that the hidden color is the same as the displayed color is clearly 2/3, as this holds if and only if the chosen card is black or white, which chooses 2 of the 3 cards. Symmetry suggests that the probability is independent of the color chosen. (This can be formalized, but requires more advanced mathematics than yet discussed.)

Experiment

Using specially constructed cards, the choice can be tested a number of times. By constructing a fraction with the denominator being the number of times “B” is on top, and the numerator being the number of times both sides are “B”, the experimenter will probably find the ratio to be near 2/3.

Note the logical fact that the B/B card contributes significantly more (in fact twice) to the number of times “B” is on top. With the card B/W there is always a 50% chance W being on top, thus in 50% of the cases card B/W is drawn, card B/W virtually does not count. Conclusively, the cards B/B and B/W are not of equal chances, because in the 50% of the cases B/W is drawn, this card is simply “disqualified”.

Members are advised of the following statistics talks taking place this week at UQ. Thanks to Ian Woods for passing this information on for the benefit of members!

Where: each in 67-641, Priestley Building (67), room 641 (6th floor), University of Queensland, St. Lucia.

A Bayesian Non-Parametric Approach to Estimating Minimum Phase Transfer Functions

 (2pm Wednesday 25/3/09)

Ross McVinish

School of Mathematics and Physics, University of Queensland

 In this seminar, I present a Bayesian nonparametric approach to the estimation of a stable, minimum phase transfer function. A minimum phase transfer function has no zeros outside the stability region, which implies that its phase component can be uniquely determined from its magnitude component, a property known as the Bode phase-magnitude relationship. The proposed approach specifies the prior for the magnitude of the transfer function and uses the Bode phase-magnitude relationship to determine the transfer function. The rate at which the posterior distribution concentrates around the true transfer function is derived. Posterior computation using Markov chain Monte Carlo simulation is briefly discussed and the approach illustrated on simulated data. For information about the speaker, see http://www.maths.uq.edu.au/MASCOS/people.html

Efficient simulation for the tail probability of a sum of the components of log-elliptical random vectors

 (10am Thursday 26/3/09)

 Leonardo Rojas Nandayapa (joint work with Soren Asmussen and José Blanchet) Department of Mathematical Sciences, Aarhus University, Denmark

Tail probabilities of the sum of the components of a Log-elliptical random vector are considered. These random vectors are obtained by an exponential transformation of random vectors with elliptical distributions – a large class of multivariate distributions which includes the multivariate normal, the multivariate-t, normal mixtures and generalized hyperbolics among others. After a brief introduction to this class of distributions, a Monte Carlo estimator for the tail probability of the sum is proposed and analyzed. In particular, for the multivariate lognormal case it is shown to have optimal theoretical properties. For information about the speaker, see http://www.thiele.au.dk/index.php?id=13&tx_thielecv_pi1[showUid]=12&cHash=01337ad499

Estimating Change-Points in Biological Sequences via the Cross-Entropy Method

(2pm Thursday 26/3/09)

Gareth Evans School of Mathematics and Physics, University of Queensland

The genomes of complex organisms, including the human genome, are known to vary in GC content along their length. That is, they vary in the local proportion of the nucleotides G and C, as opposed to the nucleotides A and T. Changes in GC content are often abrupt, producing well-defined regions. We model DNA sequences as a multiple change-point process in which the sequence is separated into segments by an unknown number of change-points, with each segment supposed to have been generated by a different process. Multiple change-point problems are important in many biological applications, particularly in the analysis of DNA sequences. Multiple change-point problems also arise in segmentation of protein sequences according to hydrophobicity. We use the Cross-Entropy method to estimate the positions of the change-points. Parameters of the process for each segment are approximated with maximum likelihood estimates. Numerical experiments illustrate the effectiveness of the approach. We obtain estimates of the locations of change-points in artificially generated sequences and compare the accuracy of these estimates with those obtained via other methods such as IsoFinder and Markov Chain Monte Carlo. Lastly, we provide examples with real data sets to illustrate the usefulness of our method. For information about the speaker, see http://www.maths.uq.edu.au/staff_visitors/academic_page.php?id=307

UQ Statistics seminars web page: http://www.maths.uq.edu.au/statistics/seminars/UQ%20Statistics%20Seminars.html

I’m also reading R.Duncan Luce’s classic “Response Times” book about reaction times, and doing fancier statistics on them. Seems like an important topic, and I have some projects in mind.