What the Fox Said to Achilles (and friends): A Graphic Story of Artificial Intelligence

 

The sum of three cubes problem and why it is interesting for artificial intelligence

The holy grail of artificial intelligence is the quest to develop general artificial general intelligence.  A considerable amount of progress has been made on specific forms of intelligence. Computers are now able to perform many tasks at suprahuman levels.  But extending these specific capabilities to more general intelligence has so far proven elusive.

One of the most important barriers  to extending computational intelligence from specific to general capabilities is an inadequate understanding of the different kinds of problems that a general intelligence system would have to solve.  Research has focused on one narrow range of problems with the apparent expectation that solving those problems will lead eventually to general problems solving.  These are problems that can be solved by parameter optimization.  Optimization means that the learning algorithm adjusts the values of its model parameters to gradually approximate the desired output of the system.  But there are problems that cannot be solved using these methods.

The sum of three cubes problem is one of these problems.  Conceptually, it is not very complicated. It could straightforwardly be solved by brute force, that is by trying numbers until a solution is found.  Still it has resisted solutions for some numbers despite more half a century of effort.
 

In general form, the three cubes problem is this: For any integer k, express that number as the sum of three integers cubed.  For example, the integer 29 can be expressed as 29 = 3³ + 1³ + 1³ (29 = 27 + 1 + 1). It is easy to determine that some numbers cannot be represented as the sum of three cubes.  For example, the number 32 cannot be expressed as the sum of three cubes, but until just recently, no one knew whether the integer 33 could be.  Is there some set of three integers that satisfy the equation 33 = x³+ y³+ z³?  In fact, until recently, there were only two numbers below 100 for which a solution was unknown, 33 and 42.  All of the others were either known to be impossible or the three integers were known.

There is no known optimization method for finding the three numbers that when cubed sum up to 33 or 42.  There are no known methods to gradually approximate a solution.  Once the correct three integers have been found, it is easy to verify that they are, in fact, correct, but there is no solution that is partially correct, only solutions that are correct or incorrect.  The best that one can do is to guess at likely numbers.  Andrew Booker, at the University of Bristol, was recently able to solve the problem for k = 33 by improving somewhat the methods used to guess potential solutions.  His method reduced the number of integers that needed to be searched by an estimated 20%, but even after this improvement, his solution consumed 23 core-years of processing time.  That is a substantial amount of effort for a fairly trivial problem.  According to Booker, “I don’t think [finding solutions to the sum of three cubes problems] are sufficiently interesting research goals in their own right to justify large amounts of money to arbitrarily hog a supercomputer.”

Why this problem is interesting for artificial intelligence

The sum of three cubes problem has resisted solution for over half a century.  This problem is very easy to describe, but difficult, or at least tedious, to solve.  Understanding the difficulty posed by this kind of problem and how that challenge was addressed is, I think, important for understanding why general intelligence is a challenge and what can be done to meet that challenge.

Current versions of machine learning can all be described in terms of three sets of numbers.  One set of numbers maps properties of the physical world to numbers that can be used by a computer.  One maps the output of the computer to properties of the physical world.  The third set of numbers represents the model that maps inputs to outputs.  Machine learning consists of adjusting this set of model numbers (using some optimization algorithm) to better approximate the desired relation between inputs and outputs.  This kind of framework can learn to recognize speech, to create novel musical creations, and play chess, go, or Jeopardy.  In fact, some version of this approach can solve any problem that can be represented in this way.

But it is still the case, that the success of these systems relies heavily on the ability of human designers to construct these three sets of numbers and select optimization algorithms to adjust the model.  The sum of three cubes problem is not amenable to an optimization approach because there is no way to determine which changes to X, Y, and Z, will bring it closer to the desired solution.  There is no way to define closer.

In 1965, I. J. Good raised the possibility of an ultraintelligent computer system that would surpass human intelligence:

“Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an 'intelligence explosion,' and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make, provided that the machine is docile enough to tell us how to keep it under control.”

Presumably, solving the sum of three cubes problem would also be among the intellectual activities that such a machine would address, since it continues to be a problem addressed by humans.  This problem is conceptually much simpler than designing intelligence programs, but it may be even less tractable. 

Booker’s improved algorithm was not discovered automatically.  There is no algorithm that we know of that can produce new algorithms like the one he produced.  It took humans over 64 years to come up with one even this good, despite fairly widespread interest in the problem.  We do not know how Booker came up with the insight leading to this new algorithm, nor do we know how to go about designing a method that could do so predictably.  General intelligence will require computers to be able to generate new problem representations and new algorithms to solve new problems, but we have little idea of how to get there.

Even this new method faced a huge combinatoric challenge.  There are just so many combinations of three numbers that could be the solution to the problem.  No matter how intelligent a system is, there ultimately may be no amount of intelligence that can eliminate this combinatoric problem.  If even the problem of finding three numbers can be combinatorically challenging, what will a general intelligence system face when trying to solve problems with even more variables?  The time required to test a large number of integers and their cubes may be reduced, but it cannot be eliminated.

To this point, no one has come up with a computer system that can design its own models.  Deep learning systems that are said to come up with their own representations actually still work by adjusting parameters in a prestructured model.  The transformations that occur within the model (moving from one layer to the next) are determined by the architecture of those layers.  For example, a linear autoencoder layer does not learn an arbitrary representation of the data, it “learns” to perform a principal component analysis, a well-known statistical technique.   So far someone still has to come up with the design of the network and optimization methods used to solve the problems.

The sum of three cubes problem could be solved by simple brute force if we were to allocate sufficient resources to its solution.  With other kinds of problems even the space in which to apply the brute-force search may be obscure.  Some insight problems, for example, are difficult until the solver finds the right representation, at which point they are typically easy to solve.  Like the sum of three cubes problem, insight problems do not admit of partial solutions that can be selected through optimization.  The key to solving is to think of the problem in the right way.  Solving these problems requires a switch in how those problems are represented. 

Here’s an insight problem whose solution may be familiar:  Curt and Goldie are lying dead on a wet rug in a locked room.  The room is in an old house near some railroad tracks.  How did they die?

Once you come up with the right model for this situation, solving it is trivial, but the difficult part is often coming up with the right representation.  There are many other insight problems, but these have not at all been studied by computer scientists so far as I am aware.  But the problem of coming up with good representations has been the very mechanism of progress in artificial intelligence.  So far it has been done slowly and painstakingly by people. 

There are many other problems that a generally intelligent system will have to address if it is ever to achieve general intelligence, let alone superintelligence.  We may someday be able to create artificial general intelligence systems that can address these problems, but it will require a different computational approach than any we have available today.  

The Singularity Called: Don't Wait Up

Dylan Azulay at emerj has just published another in a series of surveys that have been conducted over the last several years by different groups about when the technological singularity is likely to happen.  The singularity is the idea that computers will get so smart that their intelligence will grow explosively.

The notion of a technological singularity was initially proposed by Vernor Vinge in 1993, expanding on some ideas from I. J. Good and John Von Neumann.

Good wrote:

“Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an "intelligence explosion," and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make.”  Good, I. J. (1965). Speculations Concerning the First Ultraintelligent Machine, in Advances in Computers, vol 6, Franz L. Alt and Morris Rubinoff, eds., 31-88, Academic Press.

According to Vinge: “It's fair to call this event [the explosion in machine intelligence] a singularity (‘the Singularity’ for the purposes of this piece). It is a point where our old models must be discarded and a new reality rules, a point that will loom vaster and vaster over human affairs until the notion becomes a commonplace.”

The notion of the singularity combines the idea of artificial general intelligence, with the idea that such a general intelligence will be able to grow at exponential velocity.  General intelligence is a difficult enough problem, but it is solvable, I think.  But, contrary to the speculations of Good, Vinge, Bostrom, and others, it will not result in an intelligence explosion.

To understand why there will be no explosion, we can start with the 18^th Century philosophical conflict between Rationalism and Empiricism.  Simplifying somewhat, the rationalist approach assumes that the way to understanding, that is intelligence, lies principally in thinking about the world.  The empiricist approach says that understanding comes from apprehension of facts gained through experience with the world.  In order for there to be a singularity explosion, the rationalist position has be completely correct, and the empiricist position has to be completely wrong, at least so far as computational intelligence is concerned.  If all it took to achieve explosive growth in intelligence was to think about it, then the singularity would be possible, but it would leave a system lost in thought.

If understanding depends on gleaning facts from experience, then a singularity is not possible because the rate at which facts become available is not changed by increases in computational capacity.  In reality, neither pure Rationalism nor pure Empiricism is sufficient, but if we view intelligence as including the ability to solve physical world, not just virtual, problems, then a singularity of the sort Vinge discussed is simply not possible.  Computers may, indeed, increase their intelligence over time, but well designed machines and being good at designing them are not sufficient to cause an explosive expansion of intelligence.

Imagine, for example, that we could double computing capacity every few (pick one) months, days, or years.  As time goes by, the size of the increase becomes indistinguishable from vertical, and an explosion in computing capacity can be said to have occurred.  If all the computer had to do was to process symbols or mathematical values, then we might achieve a technological singularity.  The computer would think faster and faster and faster and be able to process more propositions more quickly.  Intelligence, in other words, would consist entirely of the formal problem of manipulating symbols or mathematical objects.  A computer under these conditions could become super-intelligent even if the entire universe around it somehow disappeared because it is the symbols that are important, the world is not.  But the world is important.

The board game go is conceptually very simple, but because of the number of possible moves, winning the game is challenging.  Go is a formal problem, meaning that one could play go without actually using stones or a game board, just by representing those parts symbolically or mathematically.  It is the form of the problem, not its instantiation in stones and boards that is important.

In fact, when AlphaGo played Lee Sedol, its developers did not even bother to have the computer actually place any stones on the board. Instead, the computer communicated its moves to a person who placed the stones and recorded the opponents responses.  It could have played just as well without a person placing the stones because all it really did was manipulate symbols for those stones and the board.  The physical properties of the stones and board played no role and contributed nothing to its ability to play.  The go game board and stones were merely a convenience for the humans, they played no role in the operation of the computer.

AlphaGo was trained in part by having two versions of the game play symbolically against one another. With more computer power, it could play faster and thus, theoretically learn faster.  Learning to play go is the perfect rationalist situation.  Improvement can be had just by thinking about it. No experience with a physical world is needed.  With enough computer power, its ability to play go might be seen to “explode.”

But playing go is not a good model for general intelligence.  After playing these virtual games,  it knew more because of its ability to think about the game, but intelligence in the world requires different capabilities beyond those required to play go.  Go is a formal, perfect information problem.  The two players may find it challenging to guess what the future state of the game will be following a succession of moves, but there is no uncertainty about the current state of the game.  The positions of the stones on the playing grid are perfectly known by each player.  The available moves at any point in time are perfectly known and the consequences of each move, at least the immediate consequences of that move are also perfectly known. Learning to play consisted completely of learning to predict the future consequences of each potential move.

Self-driving vehicles, in contrast, do not address a purely formal problem.  Instead, their sensors provide incomplete, faulty, information about the state of the vehicle and its surroundings.  Although some progress can be made by learning to drive a virtual simulated vehicle, there is no substitute for the feedback of driving a physical vehicle in a physical world.  Learning to drive is not a purely rationalist system. Rather it depends strongly on the system’s empirical experience with its environment. 

At least some of the problems faced by an artificial general intelligence system will be of this empiricist type.  But a self-driving vehicle that computed twice as fast, would not learn at twice the rate, because its learning depends on feedback from the world and the world does not increase its speed of providing feedback, no matter how fast the computer is. This is one of the main reasons whey there will be no intelligence explosion.  The world, not the computer, ultimately controls how fast it can learn. 

Most driving is mundane.  Nothing novel happens during most of the miles driven so there is nothing new for the computer to learn.  Unexpected events (why simulation is not enough) occur with a frequency that is entirely unrelated to the speed or capacity of the computer.  There will be no explosion in the capabilities of self-driving vehicles.  They may displace truck and taxi drivers, but they will not take over the world, and they will not do it explosively.

There are other reasons why the singularity will be a no-show.  Here is just one of them.  Expanding machine intelligence will surely require some form of machine learning.  At its most basic, machine learning is simply a method of modifying the values of certain parameters to find an optimal set of values that solve a problem.  AlphaGo was capable of learning to play go because the DeepMind team structured the computational problem in an important new way.  Self-driving cars became possible because the teams competing in the second DARPA grand challenge figured out a new way to represent the problem of driving.  Computers are great at finding optimal parameter values, but so far, they have no capability at all for figuring out how to structure problem representations so that they can be solved by finding those parameter values.

Good assumed that “the design of machines is one of these intellectual activities” just like those used to play go or drive, but he was wrong.  Structuring a problem so that a computer can find its solution is a different kind of problem that cannot be reduced to parameter value adjustment, at least  not in a timely way.  Until we can come up with appropriate methods to design solutions, artificial general intelligence will not be possible.  Albert Einstein was not known as brilliant for his ability to solve well-posed problems, rather he was renowned for his ability to design new approaches to solving certain physics problems—new theories.  Today’s computers are great at solving problems that someone has structured into equations, but none is able yet to build create new structures.  General intelligence requires this ability, and it may be achievable, but as long as general intelligence depends on empirical feedback, the chances of a technological singularity are nil.

Discriminate for fairness

As machine learning methods come to be more widely used, there is a great deal of hand-wringing about whether they produce fair results.  For example, Pro Publica reported that a widely used program intended to assess the likelihood of criminal recidivism, that is whether a person in custody would be likely to commit an additional crime, tended to over-estimate the probability that a black person would commit an additional crime and under-estimate whether a white person would.  Amazon was said to have abandoned a machine learning system that evaluated resumes for potential hires, because that program under-estimated the likely success of women and therefore, recommended against hiring them.

I don’t want to deny that these processes are biased, but I do want to try to understand why they are biased and what we can do about it.  The bias is not an inherent property of the machine learning algorithms, and we would not find its source by investigating the algorithms that go into them. 

The usual explanation is that the systems are trained on the “wrong” data and merely perpetuate the biases of the past.  If they were trained on unbiased data, the explanation goes, they would achieve less biased results.  Bias in the training data surely plays a role, but I don’t think that it is the primary explanation for the bias.

Instead, it appears that the bias comes substantially from how we approach the notion of fairness itself.  We assess fairness as if it were some property that should emerge automatically, rather than a process that must be designed in. 

What do we mean by fairness?

In the Pro Publica analysis of recidivism, the unfairness derived largely from the fact that when errors are made, they tend to be in one direction for black defendants and in the other direction for white defendants.  This bias means that black defendants are denied bail when they really do not present a risk, and white defendants are given bail when they really should remain in custody.  That bias seems to be inherently unfair, but the race of the defendant is not even considered explicitly by the program that makes this prediction. 

In the case of programs like the Amazon hiring recommendation system, fairness would seem to imply that women and men with similar histories be recommended for hiring at similar rates.  But again, the gender of the applicant is not among the factors considered explicitly by the hiring system.

Race and gender are protected factors under US law (e.g., Title VII of the Civil Rights Act of 1964).  The law states that “It shall be an unlawful employment practice for an employer … to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual’s race, color, religion, sex, or national origin.”

Although the recidivism system does not include race explicitly in its assessment, it does include such factors as whether the defendant has any family members who have ever been arrested, whether they have financial resources, etc.  As I understand it, practically every black person who might come before the court is likely to have at least one family member who has been arrested, but that is less often true for whites.  Black people are more likely than whites to be arrested, and once arrested, they are more likely than whites to be convicted and incarcerated.  Relative to their proportion in the population, they are substantially over-represented in the US prison system compared to whites.  These correlations may be the result of other biases, such as racism in the US, but they are not likely to be the result of any intentional bias being inserted into the recidivism machine learning system.  Black defendants are substantially more likely to be evaluated by the recidivism system and were more likely to be included in its training set because these same factors.  I don’t believe that anyone set out to make any of these systems biased.

The resumes written by men and women are often different.  Women tend to have more interruptions in their work history; they tend to be less assertive about seeking promotions; they use different language than men to talk about their accomplishments.  These tendencies, associated with gender are available to the system, even without any desire to impose a bias on the results.  Men are more likely to be considered for technical jobs at Amazon because they are more likely to apply for them.  Male resumes are also more likely to be used in the training set, because historically, men have filled a large majority of the technical jobs at Amazon.

One reason to be skeptical that imbalances in the training set are sufficient to explain the bias of these systems is that machine learning systems do not always learn what their designers think that they will learn.  Machine learning works by adjusting internal parameters (for example the weights of a neural network) to best realize a “mapping” from the inputs on which it is trained to the goal states that it is set.  If the system is trained to recognize cat photos versus photos of other things, it will adjust its internal parameters to most accurately achieve that result.  The system is shown a lot of labeled pictures, some of which contain cats, and some of which do not.  Modern machine learning systems are quite capable of learning distinctions like this, but there is no guarantee that they learn the same features that a person would learn.

For example, even given many thousand of training examples to classify photographs, a deep neural network system can still be “duped” into classifying a photo of a panda as a photo of a gibbon, even though both photos look to the human eye very much like a panda and not at all by a gibbon.  All it took to cause this system to misclassify the photo was to add a certain amount of apparently random visual noise to the photograph.  The misclassification of the picture when noise was added implies that the system learned features, in this case pixels, that were disrupted by the noise and not the features that a human used.

The recidivism and hiring systems, similarly, can learn to make quite accurate predictions without having to consider the same factors that a human might.  People find some features more important than others when classifying pictures.  Computers are free to choose whatever features will allow correct performance, whether a human would find them important or not.

In many cases, the features that it identifies are also applicable to other examples that it has not seen, but there is often a decrease in accuracy when a well-trained machine learning system is actually deployed by a business and applied to items (e.g., resumes) that were not drawn from the same group as the training set.  The bigger point is that for machine learning systems, the details can be more important than the overall gist and the details may be associated with the unfairness.

Simpson’s paradox and unfairness

A phenomenon related to this bias is called Simpson’s paradox, and one of the most commonly cited examples of this so-called paradox concerns the appearance of bias in the acceptance rate of men versus women to the University of California graduate school. 

The admission figures for the Berkeley campus for 1973 showed that 8442 men applied, of which 44% were accepted, and 4321 women applied, of which only 35% were accepted.  The difference between 44% and 35% acceptance is substantial and could be a violation of Title VII.

The difference in proportions would seem to indicate that the admission process was unfairly biased toward men.  But when the departments were considered individually, the results looked much different.  Graduate admission decisions are made by the individual departments, such as English, or Psychology.  The graduate school may administer the process, but it plays no role in deciding who gets in.  On deeper analysis it was found (P. J. Bickel, E. A. Hammel, J. W. O'Connell, 1975) that 6 of the 85 departments showed small bias toward admitting women and only four of them showed a small bias toward admitting men.  Although the acceptance rate for women was substantially lower than for men, individual departments were slightly more likely to favor women than men. This is the apparent paradox, departments are not biased against women, but the overall performance of the graduate school seems to be.

Rather, according to Bickel and associates, the apparent mismatch derived from the fact that women applied to different departments on average than the men did.  Women were more likely to apply to departments that had more competition for their available slots and men were more likely to apply to departments that had relatively more slots per applicant. In those days, the “hard” sciences attracted more male applicants than female, but they were also better supported with teaching assistantships and so on than the humanities departments that women were more likely to apply to. Men applied on average to departments with high rates of admission and women tended to apply to departments with low rates.  The bias in admissions was apparently not caused by the graduate school, but by the prior histories of the women, which biased them away from the hard sciences and toward the humanities.

A lot has been written about Simpson’s paradox and even whether it is a paradox at all.  The Berkeley admissions study as well as the gender bias and recidivism bias can all be explained by the correlation between a factor of interest (gender or race) and some other variable.  Graduate applications were correlated with patterns of department selection, gender bias in resume analysis is correlated with such factors as work history, language used to describe work, and so on.  Recidivism predictors are correlated with race.  Although these examples all show large discrepancies in the size of the two groups of interest (many more men applied to graduate school, many more of the defendants being considered were black rather than white, and many more the Amazon applicants were men), these differences will not disappear if all we do is add training examples. 

These systems are considered unfair, presumably because we do not think that gender or race should play a causal role in whether people are admitted, hired, or denied bail (e.g., Title VII).  Yet, gender and race are apparently correlated with factors that do affect these decisions.  Statisticians call these correlated variables confounding variables.  The way to remove them from the prediction is to treat them separately (hold them fixed).  If the ability to predict recidivism is still accurate when considering just blacks or just whites, then it may have some value.  If hiring evaluations are made for men and women separately, then there can be no unintentional bias.  Differences between men and women then, cannot explain or cause the bias because that factor is held constant for any predictions within a gender.  Women do not differ from women in general in gender-related characteristics, and so these characteristics are not able to contribute to a hiring bias toward men.

We detect unfairness by ignoring a characteristic, for example, race or gender, during the training process and then examining it during a subsequent evaluation process.  In machine learning, that is often a recipe for disaster.  Ignoring a feature during training means that that feature is uncontrolled in the result.  As a result, it would be surprising if the computer were able to produce fair results.

Hiring managers may or may not be able to ignore gender.  The evidence is pretty clear that they cannot really do it, but the US law requires that they do.  In an attempt to make these programs consistent with laws like Title VII, their designers have explicitly avoided including gender or race among the factors that are considered.  In reality, however, gender and race are still functionally present in the factors that correlate with them.  Putting a man’s name on a woman’s resume, does not make it into a male resume, but including the questions about the number of a defendant’s siblings that have been arrested does provide information about the person’s race.  The system can learn about them.  But what really causes the bias, I think, is that these factors are not included as part of the system’s goals.  

If fairness is really a goal of our machine learning system, then it should be included as a criterion by which the success of the system is judged.  Program designers leave these factors out of the evaluation because they mistakenly (in my opinion) believe that the law requires them to leave them out, but machines are unlikely to learn about them unless they are included.  I am not a lawyer, but I believe that the law concerns the outcome of the process, not the means by which that outcome is achieved.  If these factors are left out of the training evaluation, then any resemblance of a machine learning process to a fair one is entirely coincidental.  By explicitly evaluating for fairness, fairness can be achieved. That is what I think is missing from these processes. 

The goals of machine learning need not be limited to just the accuracy of a judgment.  Other criteria, including fairness can be part of the goal for which the machine learning process is being optimized.  The same kind of approach of explicitly treating factors that must be treated fairly can be used in other areas where fairness is a concern, including mapping of voting districts (gerrymandering), college admissions, and grant allocations.  Fairness can be achieved by discriminating among the factors that we use to assess fairness and including these factors directly and explicitly in our models.  By discriminating we are much more likely to achieve fairness than by leaving these factors to chance in a world where factors are not actually independent of one another.

Welcome to dead salmon data

The name for this blog was inspired by a paper by Craig M. Bennett, Abigail A. Baird, Michael B. Miller, and George L. Wolford. They studied how a dead salmon placed in a functional magnetic resonance imaging (fMRI) machine “responded” to pictures of people in emotional situations.  The salmon was not entirely forthcoming with its judgments, but Bennett and his colleagues did record some of the dead salmon’s “brain activity.” 

fMRI images are organized into tiny cubes, called “voxels.”  In flat, 2-D images, each tiny region in the image is represented by a pixel.  With the 3-D images derived using fMRI, each small 3-D region is called a voxel.  Just as the signal in a pixel represents how much light there was at that small region, the fMRI voxel represents how much biological activity there was in the corresponding 3-D region. In living brains, that activity usually represents the level of neural activity in the region.  Of the available voxels, Bennett and his colleagues found significant activity in 16 voxels, about 0.01% of the total volume. 

Based on these findings and using statistical analyses that were commonly used at the time, the authors might have concluded that they had identified the part of the fish’s brain that was responsible for making judgments about the emotional state of human beings depicted in photographs.  Before Bennett’s study was available.  About 25 – 30% of studies using fMRI published in journals used the same simple techniques as did about 80% of the papers presented at a neuroscience conference.

Obviously, there is something wrong here.  The idea of a dead fish responding to human emotion does not pass the “smell test.”  It would be remarkable enough for a live fish to identify human emotion, but it is utterly beyond reason to think that a dead one could, or that the authors were measuring brain activity in a dead fish. On top of that, we have no evidence that the fish actually engaged in the task it was assigned, as it never revealed its decisions about the emotional valence of the pictures it “observed.”

The measures of brain activity obtained using functional magnetic resonance imaging, like most other measures of just about any phenomenon include some random variation in the measured quantity. Standard statistical measures were designed to take this random variability into account.  Two measures may be numerically different without being reliable different from one another when both measures include some level of randomness. Statisticians have evolved the notion of statistical significance to indicate that an observed numerical difference is unlikely to have occurred by chance. 

But these assessments of statistical significance also involve some degree of random variability. Statisticians recognize that there are two kinds of errors that can occur.  Type I errors occur when we find a difference, when there is no actual difference (“false positive”), Type II errors occur when we fail to find a difference when there actually is one (“false negative”). 

fMRI studies compare the measured activity in each voxel during the task, with its measured level of activity during a control interval.  The more brain activity there is, the stronger the signal measured by the fMRI machine. If we were to compare the activity levels of 130,000 voxels under these two conditions, some of them would show high levels of activity just by chance because our measure is imperfect.  We would expect some percentage of these comparisons to result in false positive errors.  The specific percentage we would expect is determined by our standard of significance, often called the “p value.” It was pretty common in fMRI studies to set this value to 0.001, meaning that we would expect about 130 of the 130,000 voxels to show what appears to be a real difference, when there really is nothing but random measurement error. Multiple comparisons raise opportunities for multiple false positive errors.

There are statistical techniques for dealing with large numbers of comparisons.  In summary, these methods modify the significance standard so that differences relative to control have to be greater before one accepts that they are due to anything but chance.  When those adjustments are applied to the dead salmon fMRI data, it turns out that there are no significant areas of brain activity in the fish, as we really should expect.  When the same techniques were applied by Bennett to 11 published fMRI studies, three of them had no significant results to report at all.  If these corrections for multiple comparisons are not properly applied, researchers can come to false conclusions about the validity of their data, which can lead to wasted research effort and dead ends.

The importance of these findings extends far beyond one dead salmon, far beyond fMRI.  Spurious correlations are common throughout big-data data science.  For example, annual US spending on space, science and technology correlates over the years 1999 to 2009 with the annual incidence of suicides by hanging, strangulation, and suffocation.  http://www.tylervigen.com/spurious-correlations  If there is some kind of causal relationship here, it completely escapes me. If you have enough data and look hard enough you can find any number of correlations, but that is no guarantee that those correlations are at all meaningful.  People have a way of finding patterns, even when there are none.  Sometimes the statistics help.

Traditional statistics were developed for a time when the expensive part of research was collecting data.  Now data are plentiful, but extracting meaning from those data is still a challenge.  Simple methods are often the best, but they can also be too simple. In later posts we will consider other potential hazards of data science and what we can do to avoid them.  Unless we are careful with our data and our analyses, it is easy to get seduced into believing that we understand something when really we do not.  On the other hand, it is also easy to get seduced by approaches and methods that are popular, but may not be the best approach for the questions at hand.

In future posts, I hope to consider many of the ways in which data science, machine learning, and artificial intelligence can go wrong, but also some of the ways it can go right.  Data science is of growing importance to our businesses and even to our culture, but it is also easily misunderstood. In the same way that we see animals and faces in clouds, we may attribute to data science, particularly to artificial intelligence, properties that are not really there.  I hope to address some of these tendencies and make data science stronger and more useful as a result.

Craig Bennett,  Abigail A. Baird, Michael B. Miller, George L. Wolford (2009).  15th Annual Meeting of the Organization for Human Brain Mapping. San Francisco, CA: 2009. Neural correlates of interspecies perspective taking in the post-mortem atlantic salmon: an argument for proper multiple comparisons correction.

Craig M. Bennett, Abigail A. Baird, Michael B. Miller, George L. Wolford. (2010). Neural correlates of interspecies perspective taking in the post-mortem Atlantic Salmon: An argument for multiple comparisons correction. Journal of Serendipitous and Unexpected Results.