A Western Light

This last week samsara-news brought word of the beheading of 21 Egyptian Coptic Christians. Reasoning in such a world is a small flame. It burns bright and it cannot be extinguished. It can, however, be ignored.

The ability of humans to reason is Promethean. This is the sentence that started this series of posts exploring the human ability to reason. Prometheus stole fire from the gods and gave it to mankind that we might find warmth and comfort in the great dark of the silent, star filled night. For his gift Prometheus was condemned to eternal torture by Zeus; forever chained to a rock while day in and day out an eagle devours his liver. The eagle with its razor sharp sight is said to feast on the organ that cleans the blood. In the twilight language of symbolism and myth this situates the classical etymology of Prometheus as “forethought” as that which is associated with life itself, for the blood is life. The ancient myth does not shy away from teaching that forethought while allowing us to see far is also that which brings suffering. Who would deny it is so among those who have studied the ongoing ecocides? Looking into the future and seeing a die-off due to the ecological ignorance of our society is quite exquisitely depressing.

The difficulty of such a fundamental shift in the assessment of one of society’s most fundamental cornerstones should not be underestimated. That careful reasoning is now delivering news about a horror filled future is fueling a resurgence in the irrational. It is not hard to sense that there is an ill wind blowing through our days of late. It is not hard to sense where the fault lines are hiding or to sense their rumblings. I am reminded of Carl Jung’s premonition of the start of WWI.

The modern world inherits many of its defining characteristics from the Age of the Enlightenment when human reasoning was held up as the road less traveled, the one trustworthy guidepost through the thickets of superstition, religious fanaticism and its accompanying endless warfare. Thinkers dared to dream of a day in which universal education would spread the light of the enlightenment across all the classes of society. The greatest artistic achievements and the loftiest philosophies were thought to soon enrich the lives of the working poor as much as they had the non-working rich. Governments would no longer be ruled by religious institutions and the vagaries of dogmatic disputes but instead would create a secular space in which all peoples could live and worship as they see fit.

All of this is such obvious history we in the west rarely take the time to appreciate the uniqueness of our inheritance. For my part the non-theist position of Buddhism respects this hard earned position. The Dali Lama has taught for years now that there is a need for a secular ethics, an ethics based in reasoning and not revelation. How hard is it really to work carefully through the ethical questions each of us are confronted with? All the religions of the world are in basic agreement about what it means to live a good human life. They teach that one should not lie, steal, kill, use reproductive urges to harm others, and to not become so intoxicated that you forget the first four. Every one of these items can be defended on the basis of human dignity without the need to have recourse to any potentially divisive religious assertions.

Where the dark impulses rule, this simple agenda of grounding an ethical life in reasonable considerations comes to seem too restrictive. In bloodlust comes the hunger to hurt others, to make them pay for the hurt you suffer, to force your will upon their flesh and gain, for a moment, a sense that you are in control. The simple ethical principles grounded in reason cannot be twisted to justify murder and mayhem, so societies the world over and throughout history have succumbed to temptations to believe those pied pipers that pipe the tune the people want to hear: you alone are sacred and special and the other threatens you: “God says kill them.”

Is this not the other face of religion? One of the many virtues of the Bible is how blatant this other face of religion is described, with a jealous and angry god exterminating one society after another. The same is found on the battlefields of the Bhagavad Gita, in the stories of the Greek’s Ares and the Roman’s Mars and of course in the tales of Mohammed and the Jihad as seen in the news last week. The history of the last century puts paid to any hope that secular movements would be protected from such deviations. The Communist gulag and the Nazi death camps made clear that these eruptions of collective psychosis do not need explicitly religious breeding grounds.

Now that we have a model for what reasoning actually is, it is possible to find a nuanced position for this uniquely powerful cognitive capacity. It is not a savior nor is it a beguiling devil in disguise. Reason is an unshakable witness to what is real and true about being human in this world, a presence deep within the body-mind of what is real and true and what is not. As has been mentioned before it is not something you can fool or force to behave as you would wish but retains a degree of autonomy. Life lives us. The full experience of the “really real” is bigger than reason alone but it does not contradict reason, which is the beauty of it.

Those ethical guidelines known the world over? Those are the teachings that naturally arise when one looks upon other life forms with compassion. Compassion is the most rational response to a completely interdependent world full of unique, impermanent yet sensate beings. Why? Because it entails what is most powerful about our experience. The most intimate knowing you have of the world will be found in the interface between your feelings and the thoughts that accompany them as they encounter the world. Assuming the same experience, with all its emotional impact, is what the person in front of you also experiences is the sign of human maturity. We call those who cannot understand others to be as legitimately conscious and sensitive as themselves psychopaths and sociopaths. They lack what cognitive science call the theory of mind; the working hypothesis that others minds are the same as one’s own. Acts of violence, in war or otherwise, all share a fundamental disregard for this ground of being.

Indulge me for a moment if you would and allow a few overly simplistic generalizations. In eastern cultures there has been a tendency to dismiss the power of reasoning a bit too easily. There is an iconoclastic bursting forth of paradoxes we find refreshing but in less skilled hands has led to tossing reason out instead of carefully laying it down before that which is bigger. Generally the eastern cultures have valued the aesthetic sense with art and ritual playing important roles; they are not hounded by the Faustian hunger to know we see characterizing western cultures. The western world has gone the other extreme and dismissed dreams, mysticisms and all other manifestations of mind except those that can be reduced to reasoning about measurable properties as without value, meaning or importance. They are without value for the workplace, without meaning for discussions among the scientific intelligentsia and without importance within our theologies.

Properly appraising the ability to reason must lie somewhere between these two extremes. This is what the model we have been discussing these last few posts helps me with. It is reason that guides us when we discover how to build a bridge that will not collapse or how washing your hands before performing surgery benefits the patient. There are countless daily tasks necessary for survival and wellbeing, each of which is made more effective and powerful due to having been carefully thought about and addressed with an artifact of ingenuity. Many of our taken for granted features of the modern environment are the result of centuries of careful reasonings that were built up generation after generation. We should appreciate just how much hard work goes into engineering. The slide rules are often unruly, the elegant algorithms lose their shine in the real code running all our devices and the tolerances for error have become so small meeting their strictures is an unending chore.  To be awake to the reality of our world as it really is today needs to include this understanding so we can remain grateful for the infrastructure we enjoy and the many benefits of technology, even while we remain critical of it.

On the other hand reasoning is notoriously unable to deliver the goods when the questions being asked are about existential meaning; the ultimate purpose of the love and suffering experienced by our human nervous systems. Scientific explanations of depression do not lift depression. Scientific explanations of evolution do not comfort a bereaved widow.

For these features of our human experience the arts are more appropriate. There are reasons the heart knows that reason is not aware of, to paraphrase Pascal. The passionate embrace of lovers is captured, somewhat, in the alternating swelling and gentle breezes of musical expression, the grief of the lovers parted by death is addressed through the tragedies of the stage and screen more directly than by a research paper on molecular biology. In the beauty of form and color the sculpture’s gifts bring us a grace of understanding that reaches a place of feelings running much deeper than the calculations of reasoning – running into those places where blood is thicker than water, where the painful feasting on the liver continues.

The foresight reason grants us is couched in probability. Shrouded in uncertainty the future remains ever new. Still, we are not blind; we grope our way forward with the light of evidence. Reason allows us to be sure of something, as sure as we can be. Ask an engineer what pressure a given piece of steel can withstand and they will answer with a high degree of confidence, plus or minus a bit of course. In the world of feel-good mass entertainment and the lax, anything-goes cultural milieu it nurtures, it is important not to lose sight of this ability of careful reasoning to grant us a high degree of confidence. It aids the contemplative to maintain their individual diligence against the madness around them. It also aids cutting through the BS and the wishy-washy smoke screens deliberately created around the ecological truth of our time.

The Probability of a Proposition

I would like to arm my readers with the ability to quickly recognize invalid inferences. Inference will be explored below but first I want to say a few words about how careful thinking is the best medicine for of our times. My working position is that we cannot deal with the problems of the day with the same thinking that created them and that the pollutions, extinctions and other abuses of our environment reflect states of cultural or inner consciousness that are equally ill.

Advertising and PR find endless ways to twist the human heart and mind into painful contortions from which they can force people to act to relieve the psychic pressure – to buy this or that. We fill our heads with non-stop lying images. These fields started as information rich attempts to persuade potential buyers by sharing the virtues of the products in question. Does that seem incredible? Take a look at this typical ad from the age that saw the birth of advertising.

searsjuanita1911It was not long before the ad men (and they were men for the most part) discovered it was much more effective to bypass the reasoning mind and directly manipulating the hopes and fears we all entertain around death, sex, status and social insecurities were much more effective. One thing led to another until today we are literally awash in invalid inferences.

When Subaru says its car is love and fills its commercial messages with images of young couples in non-polluted, idyllic nature scenes or warm family relations the logic involved runs something like this:

You feel bad, as you should because you do not own our car.
Here is an example of people happy with their lives because they have love in them.
You need to own our car if you want to have love in your life.

Or a politician’s campaign:

You feel bad, as you should since I am not in office.
Here’s an example of what the guy in office does wrong.
You need to vote for me if you want to feel better about your self, your country and your future.

Everywhere the first move is to create dissatisfaction, a lack of contentment with what you already experience in the real world. It is business 101 – create the need and sell it. This is why so much of what you see and hear is surreal, animated; camera tricks being used to create worlds impossible to realize in reality. They maximize the contrast with your daily experience of people, nature, emotions and social interactions. If you have a TV watch it without the sound on for a few hours. Watch with your B.S. detector well fed, rested and ready for action. What’s real?

By the way, thinking about doing the little experiment and actually doing it with the laboratory of your own nervous system are two very different things. The whole point is to reclaim the ability to have experiences for your self and not be satisfied with the canned goods being offered. Learning is by doing so the whole body-mind is involved. Living vicariously is a trap, unworthy of the opportunity we receive with one more precious day to be alive.

If you dismiss these concerns about the medium being the message with the line they taught you – ‘Oh, I know it is not real, just an entertaining fantasy’ – then you are naive about how the world really works. The for-profit companies creating these fantasies would not continue to invest billions of dollars if the product did not induce people to act as those companies desire. Reams of unpopular research over decades supports the claim that these techniques successfully manipulate public behavior.

There are a number of reasons you might want to consider developing cognitive Aikido for dealing with these things. For one, as the slow grind of social collapse continues under the weight of diminishing energy resources and increasing pollutions, fewer and fewer of us will be found in the inner circle of winners where the consumer paradise is rumored to be found. Not that the number of images of people supposedly enjoying that tin-foil paradise will lessen, far from it. Put bluntly, it is just going to drive you crazy if you believe on some level your happiness depends on purchases you cannot afford. Another good argument for learning effective counter-measures is that peace of mind and contentment strengthen the immune system. In a world where antibiotics and healthcare systems increasingly fail to deliver it could be having inner contentment  is our most practical avenue to a long and healthy life, at least for those of us outside of the charmed circle. The final reason I will ask you to consider is simply what affect your life will have on others. Everywhere in hyper-capitalism’s twilight people are over-worked and under-appreciated. Simple common courtesy is not common, the happy-to-be-alive radiance once seen on people’s faces has all but disappeared and the crushing burden of our secular guilt due to our ongoing participation in ecocide have cast a darkness over the developed world which no one fully escapes. Use this cognitive Aikido to fight back and not only will you become less susceptible to those who would manipulate your most intimate being, but you just might become the glimmer of light that those around you need to get through their day without becoming monsters.

Without further ado on to another excerpt from my book project.

Non-Bayesian statistics and probability has mostly been a study of how chance and randomness affect events. In this approach it makes sense to talk about the probability that event X will occur but it does not make sense to talk about the probability of a proposition. Bayesian thought is nothing less than a reconsideration of these fundamental definitions. It finds that probability encompasses statistics once it is given a proper theoretical foundation. This new foundation builds on the use of probability as a guide to reasoning under uncertain evidence. It is easy to spell out the differences by quickly reviewing the basics of logic. Here we come to the heart of the matter of this Bayesian conceptual revolution. Logic as expressed in the predicate calculus is highly technical. This presentation is only as technical as needed to share a sense of the conceptual coherence this alternative view of probability provides.

Here are the classic formations of proper deductive logic side by side with a typical application. These diagrams follow the standard logical presentation in which a line separates the premises of the argument from the conclusion; the line represents “therefore”.

Logic1These illustrate the extent to which proper deductive conclusions can be drawn. Each alternative not listed leads to logical fallacies if the scope of logic remains deductive. A is referred to as the antecedent, B as the consequence. Each can be assigned what is referred to as a truth value. In deductive logic the only truth values allowed are true and false and so we talk of valid and invalid arguments. If one attempts to reason that B is true therefore A is true one commits the logical fallacy of affirming the consequence. For example if Bob did not study then he fails the test – he failed the test – therefore Bob did not study. It is easy to see that this incorrectly removes all the other reasons for which Bob may not have passed the test; he simply couldn’t understand the material, was too ill to attend class on the day of the test, etc.

If one tries to reason by drawing a conclusion from A being false therefore B is false one commits the logical fallacy of denying the antecedent. Staying with Bob such invalid reasoning runs along these lines; if Bob studied then he will pass the test – he did not study – therefore he did not pass the test. Again it is easy to see that this does not take into account all the other possible reasons Bob might have been able to pass the test; he had already learned the material, cheated, got lucky, etc.

Here is the rub. These very fallacies are often the only form of reasoning available for considering questions in the real world. We humans confront them multiple times a day. A weaker form of syllogism is possible if one extends the scope of the possible truth values from only true and false being allowed so that they are able to take on a range of probabilities. Now instead of logical fallacies it becomes a question of correctly reasoning about uncertainties, inductive logic. We say an inductive argument is weak or strong.

Logic2Notice that these are entirely logical connections. The relationship illustrated in the weaker syllogisms is not in the direction of cause and effect; it does not assert that because there are clouds there will be rain, after all many times in the past it has been cloudy but has not rained. Instead the direction is one of logic; it asserts that if there is rain then there must also be clouds. If it is 9 am and one is trying to decide to take an umbrella or not the state of the sky and past experience are the evidence one has on which to form a conclusion. You form a prediction, one of many you will attempt throughout the day. To make the prediction you weigh the evidence, you inductively consider just how cloudy is it, how dark are the clouds, what way is the wind blowing, and what did they forecast on the news last night?

This extension to plausible inference was examined by G. Polya in ‘Mathematics and Plausible Reasoning’ (1954) particularly volume two subtitled appropriately ‘Patterns of Plausible Inference.’ Here the weak syllogisms are taken one more step where they are shown to explain other typical characteristics of inferences. We all seem to intuitively understand “the verification of certain consequences strengthens our belief more and that of others strengthens it less” (Polya pg. 6). We also operate as if further substantiating evidence increases plausibility though sometimes the new evidence affects the strength of weakness of our conclusions considerably and sometimes only slightly.

Logic3There are many more forms of logical patterns in both inductive and deductive logic. There are patterns for dealing with propositions that are incompatible with each other and patterns for adding the quantifiers “some” and “all” to the propositions.  These few provide sufficient illustration for our purposes. It is reassuring that the extension of logic by using plausibility agrees with our common sense notions but is there anything more substantial to bring in their defense? Indeed there is. The physicist Richard Cox was able to derive the laws of probability from a set of postulates that justifies the logical interpretation of probability. He does so using Boolean algebra. Regular algebra deals with quantities; Boolean algebra deals with propositions. Boolean algebra defines operators on which today’s computers rely, for example, the logical operations indicated by AND, OR, NOT. In “The Algebra of Probable Inference” (Cox 1961) Cox derives probability theory from an extension of Boolean algebra and in so doing proves it is the only theory of inductive reasoning that maintains logical consistency. It is said that Cox returned probability theory to its original 18th century roots as formalized by Laplace. He does this by proving probability theory to be the axioms of logic when logic deals with uncertainty. In the 1950s this approach was considered too subjective to be used in science and engineering so the alternative foundation of probability theory in the frequentist school was developed. Cox’s brilliant achievement was to show that the original logical foundation was in fact the most coherent. The Cox Theorem has provided a rigorous mathematical mapping of inductive logic to probability theory. It is a fundamental intellectual achievement.

One, I might add, that I believe deserves to be shepherded through the coming societal upheavals.

Oh, and Subaru is not love, it is a car. One of an estimated 900 million that are currently running around on our planet, eating our future alive. They are saying there will be over two billion light vehicles on the road by 2050. Really? Inference is how we are able to think about the future. Right now there is no more important ethical and practical question to be asking. Like it or not the ecological evidence is overwhelmingly telling us the coming generations will suffer under some catastrophic tipping point or another. That is the most probable outcome. Business as usual will not continue and whatever comes next will be cleaning up or avoiding the toxic results of our actions for centuries. Time to wake up and smell the burnt toast, anyone noticed any strange weather here in the U.S. of late? Two billion cars by 2050 is, to put it as politely as I can, crazy talk.

The data matters, that the prior convictions are reasonable matters, that we understand the best we can do is sketch a probability curve of what is most likely matters. It is the truth of the strength of human knowledge. Epistemologically these three curves will meld and mold each other into a most probable outcome, the result of our careful reasoning. Carrying the weight of our best understanding we are ethically obligated to act as if it is the truth, even though our result is also a curve and will meet new data and evolve through another cycle.

The point of all this is this: the data that has been gathered in the last few years concerning the severity of the ecological crises has exceed what was expected by most of our models, sometimes considerably. Some of the weather events outside your door were not expected for another decade or more. The models are being recalibrated but it should be common knowledge that in the conservative IPCC reports of 2007 the worst scenarios are the ones the historic data matched. It should be common knowledge that the same historic trend lines matching the worst scenarios modeled is also the case for the Limits to Growth studies. All of these are saying that what we are doing now is almost certainly stealing our ease at the cost of massive suffering by people just like us who happen to be born in the future instead of now. Or, as a Buddhist I cannot help but wonder, might those future beings be us in some way, paying the piper as cause and effect work their way down the centuries of DNA time? Either way the here and now is real, precious and threatened.

Flip You For It

“Reality is that which, when you stop believing in it, doesn’t go away”.
How to Build a Universe That Doesn’t Fall Apart Two Days Later, Philip K. Dick

 

Last week’s cliff hanger flashed Bayes Theorem and today we are going to put it through its paces. I’m repeating the same equation here in words( | is the symbol for given, ∝ is the symbol for is proportional to):

Probability (hypothesis|data) ∝ Probability (data|hypothesis) * Probability (hypothesis)

BayesEquationWordedRemember the point of the equation is to weigh a belief in light of some new data. This new degree of belief in the hypothesis is called the posterior probability, what we have as a result of the operations. The operations ask how likely is this data given our belief and just how probable is that belief anyway? The second form normalizes the numbers by dividing by the evidence so that the distributions that they represent sum to 1.

The posterior then becomes your prior degree of belief the next time new data confronts your hypothesis. Iterating the process, chaining the equation to itself this way, is one way of modeling the human reasoning process. An example pertinent to the concerns of this blog: I believe anthropomorphic climate change is really real.

The process I went through to arrive at this conclusion went something along these lines. First being open-minded I had no idea what to believe, nothing that would satisfy my most critical searches for evidence justifying a position one way or the other. I do know the earth is warming, a trend both sides mostly agree on. I understand how green house gas physics is a part of what has allowed the biosphere to flourish for billions of years and how changes in its composition are linked to changes in temperature throughout geological periods. These are some of the relevant prior understandings I am bringing to the question. I do not know what to make of the claim that human activity is having a significant effect on these gases. This is the state of maximum entropy in the lingo of Bayes. Then I study the data about the melting ice caps; say a dozen peer reviewed articles, handful of books, a couple of documentaries, and plenty of photographic evidence. The majority of the evidence is arguing that the rate of melting is accelerating because the contributions of industrial gases are a statistically significant factor. Now my honest sense of what is real finds veracity in the claim that climate change is related to human activity. Say I became 70% convinced; there is probably something to the claim that anthropomorphic climate change is really real. Now when I turn to the study of ocean acidification I bring my previously reasoned position with me. As the process of my study continued through all the types of data available and how the models built of the data are constructed and interpreted eventually the claim worked its way into that inner bucket of “this is real.”

Much money is spent and enormous efforts are applied to the public conversations around climate change to present the public with the impression that either side might be right. A concerted effort by those with much to lose has created the impression that the question is 50 / 50; maybe yes, maybe no. Big oil and their bought talking-heads in congress and on TV would like to say to our children’s generation – “I’ll flip you for it.” A more indefensible position is hard to imagine. The circumstances are complex enough to rule out any simple black and white conclusions – yes, human activity is causing all climate change or no, human activity has absolutely nothing to do with climate change. This is not a situation in which a binary choice applies as if flipping a coin. We will look at flipping a coin in just a moment to draw the contrast in as stark of terms as I am capable of.

If we are going to find any peace of mind we are going to need to learn to think straight and one of the indispensable skills that requires is fine tuning our B.S. detectors.

Now to debate my position is always welcome. Intellectual honesty and integrity lay down a simple rule: bring me data and / or an alternative hypothesis that will convince me otherwise. There is a place for poetics, rhetoric, spin and color. This is not that place. This is reasoning and it is being applied to life and death questions. As we are all in this together I think we must agree that reason is the only reliable guide we seem to have access to as owners of a finite understanding embedded in the universe we are reasoning about. Remember that bit about being able to measure also providing access to the only objectivity we can claim? It is the same type of thing here. You can assert without further evidence that a man in the clouds or a deceased uncle told you the claim was a lie, but I can hardly be slighted for dismissing you as not sufficiently serious given the stakes. You can bring out data but on this question the overwhelming majority in every relevant field is against you. Going against the objectivity of the majority is indeed your prerogative, after all, absolutes are off the table and how else will the paradigms change? Still, to assert your position is anything but one of the inhabitants of cranksville, that would simply be dishonest. What my mother would call a lie.

So far I have just pulled percentages out of the air in my examples. The actual process is hardly so arbitrary; in fact it is in the transparency of the reasoning mechanic that the great strength of Bayesian work shines. It is why science is using it in more and more of its modeling, why spy agencies have been using it for ages and those building models of the brain find it central to their work. If you want to join the fun and games I’d like to mention and thank Mr. Kruschke for his fine guide, Doing Bayesian Data Analysis.

What follows is an excerpt from a project around Bayesian thought. It is offered to convey some sense of the processes.

There are two main characteristics of probability distributions to keep in mind: the area under the curve always sums to 1 and the shapes of the curves shift to where the bulk of the probability is to be found. Using various shapes allows us to express our degrees of belief be they small, large or indifferent. This illustration uses the Beta distribution as a convenient way to express degrees of beliefs though there are many others and there is no mathematical requirement for the prior to be expressible as a function at all. As long as the curve’s area sums to one, any conceivable shape can be drawn point by point using a grid approach. Let’s return to our investigation of that strange animal in our distribution jungle, the prior. It combines with the data in the likelihood, the crucible of the equation where their interaction results in an updated belief. For the likelihood in what follows a binomial distribution is used to model a binary outcome. We will graphically explore what happens as a prior encounters data that comes in all shapes and sizes. Some data we encounter is expected while other data catches us by surprise.

A friend has given you a shiny half dollar for your birthday. He assures you that this is a very special coin and encourages you to flip it to see if it comes up heads or tails. You look it over carefully but see nothing amiss so you expect the chance of the coin coming up heads to be about 50 – 50. You give the coin a good flip and it lands tails. Three more flips all come up tails too. You notice a mischievous smirk on your friends face but throwing caution to the wind flip the coin a fourth time and now it lands heads up. Five more tosses all come up tails. “I’ll bet you $50 the next flip comes up tails” your friend offers. Taken aback you begin to ask yourself just exactly how much you believe that this is indeed a fair half dollar.

I will not keep you in suspense. The coin was purchased at a magic shop where the dealer assured your friend that it was specially manufactured with a bias for landing tail side up. Let’s see how a Bayesian model deals with this situation. A fair coin can be modeled as having a 0.5 bias, meaning it has an equal chance of coming up heads or tails. Because your friend is smart enough not to believe magicians, he tried the coin out in the store before the purchase. This being a magic store his prior belief about the fairness of the coin was completely uncommitted, far as he was concerned it might have any bias at all. In the graph below this uncommitted prior belief is modeled as a Beta(1,1) distribution, a straight line covering an area summing to 1 illustrating all outcomes are considered equally likely. The likelihood below the prior shows your friend flipped the coin ten times in the magic store and only once did it come up heads. The posterior in the lowermost panel mirrors the likelihood, the Beta(1,1) prior having no affect. In the bottom panel the bias for heads is shown as .1. The center of the distribution peak convinced your friend the coin had bias that would cause it to land heads up only one tenth of the time. The deal with the proprietor of the magic store was consummated and now here you are wondering about the innocence of the very same coin.

BayesCoin1This is not the first coin you have ever seen, you’ve been around the block. You consider that of all the coins you have encountered they seemed to flip fair, maybe not perfectly unbiased but for the most part trustable enough to decide which football team should go first. You have a prior with most of the area around 0.5 but are willing to account for some variations. The Beta(2,2) distribution used below expresses this nicely. The first column of graphs show how your prior belief in a distribution centered about 0.5 would change to one closer to 0.2 if you also flipped the coin ten times and observed heads only once. Notice that the prior has had its influence; you are not willing to grant the bias is one tenth on just ten observations. So far in our story you have actually flipped the coin nine times, do you take the bet? Just how many tosses would it take to overcome the effect of your prior expectation so that you also arrive at the correct estimate of the bias centered at one tenth? The second column shows it would take about 180 throws. You could be playing with your new birthday present a long time.

BayesCoin2What if your prior conviction of the fairness of the coin was even stronger? You reason that half dollars are minted by the U.S. government according to strict specifications. Every coin may not be exactly 100% unbiased but surely if there is a bit of bias it is small. A Beta(10,10) prior captures your considerations and the graphs below tell the story. Even with only one head showing up in ten tosses the resulting degree of belief in this half dollar’s bias against heads is only about 0.35. You still expect to see three or four heads in ten throws. Lucky for you, though you trust the government mints you long ago learned your friend can be sneaky. You decline the bet and go off to dinner together, paying dutch. This is a fairytale ending, a very good thing. How many more throws would it have taken to bring your posterior belief around to one tenth when starting with this stronger prior? A whopping 1,600 throws, you could still be flipping that half dollar next year when your friend came over to give you a deck of cards for your birthday…

BayesCoin3I hope this illustrated how the data matters. Honest interpretations of it are possible because of the different understandings each of us develops over the course of our life experiences and studies. What will convince one person will not necessarily convince another and not just because they are refusing to reason with care. All this is captured in this simple example of the Bayesian explanation of reasoning but it is impossible to miss the larger implications; given sufficient evidence all people, regardless of their prior convictions, will tend towards the same inescapably probabilistic conclusions.

This sort of reasoning is a public affair. It is the Lingua Franca of social conversations that are involved with contingency planning, risk analysis and a whole host of other critical processes. The transparency of assigning probability allows us to evaluate each others positions. It is an adult form of conversation for adult issues that will directly affect the degree of suffering occurring in our world.

Hypothesis and Evidence

“A wise man proportions his belief to the evidence.”
An Enquiry Concerning Human Understanding, David Hume 1748

 

I would like to thank all those readers who have stuck with me through this investigation of reasoning. This is not as passion filled as a tirade against our blood-soaked entertainment or a timely warning and threat from the news of big oil and the Saudis filling our papers of late. This is looking to share something a bit more substantial, a way of understanding that might make it a bit easier to cope with the daily dose of foolishness and bad news.

Sometimes I have been able to restore compassion by using a poetic model of my fellow creatures that leaves aside the ethical sinner – saint dichotomy and dares to look a little deeper into the physiological substrate on which all human experience depends. In this model we are more like robots with a few circuits mis-wired due to traumas of every imaginable stripe. Not just robots, we are far too creative and unpredictable for that. We are robots holding wounded angels as carefully as we can, like the image I saw on a cover of a book long ago: ATAs we enthroned our machines we came to resemble them. I like this image, almost an icon of the fossil fueled industrial age. It captures where we find ourselves, about two steps from chapel perilous. Ruled by cultural values from the so-called age of reason it is important we understand just what this reasoning is before we are prepared to really get what the teachers mean by saying the mind is in the heart center and that it is the body that is centered in the skull.

Moving through the world, what is it that we humans are actually doing? We have (or are?) a nervous system hooked up to senses and a brain. It is relaying information in the form of electrical signals constantly. Some of these signals might run into the brain, others will inform glands, muscles, and a whole host of biological processes in a never ending quest to maintain homeostasis and coherency. Of those that make it to the brain a few might make it through numerous cascades of neural nets from the more primitive layers to the neocortex where our conscious lives take place, at least for the most part. Numerous poisons and handicaps can cripple this process, thwarting its proper functioning anywhere along the way from the senses to the visceral tissues involved all the way up to the brain itself. What are we going to do? This is what we have to work with.

While the pathologies are important, more so than our culture enamored as it is with health, youth and beauty may ever understand, it is not the focus of today’s post. Assume that the signals are arriving without incident, that the information they carry is delivered accurately as possible and that the reception of the electrical signals in the wetware brain’s neural nets is weighted and organized optimally. What is the brain doing with all that information?

I propose that it is making models of its experience of the world. By laying down tracks among the nets memories are being formed and reformed. These nets take the raw data and categorize and classify it in a multitude of ways. Cognitive science has been able to use imaging to confirm what we all experience; simple concepts are used to build more complex concepts which in turn are used to form even more complex thoughts along a hierarchy of emergent insights. These ladders of insights are not necessarily correct or accurate, though they tend to feel as if they are in their moments of coming together. The process of thinking is experienced as an ongoing, piecemeal, additive function energized by a sense of expanding insight. It feels like we are really figuring something out, that what was formerly vague is becoming more clear, what was formerly confusing is making more sense.

Sometimes these ladders of emergent discoveries survive the cold light of rational analysis on a Monday morning, sometimes they do not. Evolution evidently designed our brains to be these kinds of information processing machines. There is survival value in the ability to take the raw data from the inner and outer environments as conveyed through the nervous system as information channel and turn them into organized pictures of what we take to be really real. Despite how it might feel when we are soaring through the inner skies of learning, if I may paint a poetic picture of the process, the feeling is no guarantee that the actual cognitions are valid.

The survival value of this cognitive ability comes from its influence on how we make decisions about what actions we will pursue. How we react to circumstances and how we choose what is worthwhile to work and strive for are all colored by the models we have made about what the world is and what we are within it. With action comes risk. With action comes the chance of failure. With action comes the long arm of the law of cause and effect. Choosing not to act, is an act. It is inescapable.

The other inescapable fact of our existential situation is that all of this takes place in an environment of uncertainty. The building of our models, the precepts our senses first create, the noisy information channels and the specific causes and conditions accompanying a particular action are all thoroughly surrounded by uncertainty. Remember the robot and the angel?  A recognition of the basic physiological substrate and the nature of our cognitive power, which is always and everywhere embedded in one environment or another, leads me to conclude all our beliefs are of the nature of hypothesis. They are tentative, subject to alteration as needed under the influence of new evidence. Belief is how we experience the power of the evidence we have reasoned about.

Most of the evidence we encounter will consist of data that is what we expected. After all we have built our models from the gathering of prior experiences and fashioned it to capture what those experiences have taught us to expect is most likely. Such data is said to have low information content. Gregory Bateson identified information as the difference that makes a difference. It is measured by the amount of uncertainty that it removes. See how all these elements are starting to come together?

We see a man drop an apple and it hits the concrete sidewalk; the outcome is thoroughly expected and our model of how things move when dropped is barely affected. Maybe it is strengthened a little but the amount of learning is minimal. On the other hand if he drops the apple and it flies up into the air then we are shocked, surprised. Now there is a crisis of sorts in the stability of our model. We wonder what the Dickens is going on. Magicians use this feature of our mental makeup all the time to deliver surprises and the unexpected.

All these elements and their relationships to learning are expressed in the incredibly conscience language of mathematics in what is known as conditional probability or Bayes theorem. Bayes theorem is used to determine how likely the proposed hypothesis is given the evidence. It transforms the prior probability into what is referred to as the posterior probability. In symbols it calculates p(hypothesis |evidence) which is read “the probability of the hypothesis given the evidence.” Consider the implications of those italicized words. It captures all rational striving for human understanding.

Bayes Theorem looks like this in which p( ) is the probability, H the hypothesis, E the evidence:eq_bayesp(H) is the prior probability that H is correct before taking into account the current evidence E.

p(E | H) is the conditional probability of seeing the evidence E given that the hypothesis H is true, often called the likelihood.

p(E) is the marginal probability of the evidence, how likely this particular evidence is without respect to the current hypothesis or under the condition of any possible hypothesis.

p(H | E) is the posterior probability, the result. It provides the probability that the hypothesis is true given the evidence and the previous belief in the hypothesis.

Next week we are going to use this machinery to graphically explore what happens as a prior encounters data that comes in all shapes and sizes. Some of the data we encounter will be expected while other data will catch us by surprise.

Measure for Measure

Our scientific understanding is powerful to the degree that it corresponds with the actual world of experience. The degree to which our understanding achieves this fit is judged by practical considerations; can we use it to predict outcomes of our investigations into what remains unknown or use it to control events in the real world? Understanding increases as we work to tune this correspondence between these bodies of knowledge and the actual structural, energetic and information patterns in the universe. Our rockets get where they’re going, our rounds fire straight.

Today we bear the fruit of three or four centuries of methodological investigation of how the things we perceive around us behave. This body of knowledge comes packaged in the form of conceptual models having, more often than not, a mathematical expression. The math holds the unique philosophy of this whole endeavor. These models are built on the ability to measure something. By measuring human beings are able to transcend their subjectivity and achieve a precision that is readily communicable. The objectivity, for all the flack it has run into in modern critiques of the philosophy of the scientific method, remains very real.

Remember the different types of yes and no we looked at last week? This is similar. One person says the stick is short, another that it is of medium length. Each is honestly reporting what they experience from within the inner jungle of their prior contexts. The power of the maths is seen when they both agree the stick measures 11 inches.

So measuring things has this useful characteristic; an ability to demonstrate an aspect of things which can be agreed on by anyone of sound mind and body. The assertion that a stick is 11 inches long is quickly verified or falsified by anyone with a ruler, in any country at any time regardless of their political or religious beliefs, the weather outside or an infinity of other variables. Access to the stick in question and a ruler marked with the agreed upon (yet ultimately arbitrary) metric is all that is required.

It would not be off the mark to explain these last few centuries of scientific exploration as an ever more extensive and subtle scramble to learn how to measure the mysterious events that surround us. Eventually Chevelier de Mere, after a particularly bruising loss dicing with friends, wondered if there might be a way to measure the seemingly random. Asking his mathematically inclined friend Blaise Pascal to look at the problem sparked a fire that started the new branches of mathematics we today associate most closely with science; probability and statistics. Try to imagine the first time something as seemingly random as tossing a pair of dice began to show its generalized behavior; that it was not random in the aggregate, only in the individual throws.

Here was something new. Not a measurement that could be confirmed by someone else with a single reading but one that required reproducing a series of events. Additionally in any given series the actual outcome might differ from the predicted one but over enough trials the pattern emerges. Everything about this type of metric made its proper use, and properly understanding it, a bit tricky. Today we manage to work with these probabilities very effectively through the use of confidence intervals and margins of error. Probability is not as easy to use as a ruler but is just as objective and precise in its own way.

This act of measuring things can become surprisingly complex. The length of a shoreline depends on the scale of ‘ruggedness’ you choose, as Mandelbrot taught us. Length itself changes under relativistic conditions. But these are dwarfed by a more basic fact about measurements as they occur in the real world; few are in perfect accord with theoretical predictions. They are close enough, which is well defined, and this is good enough. It has to be, it is all we have to work with.

For example let us assume an experiment in electronic circuits. We are to measure the resistance in the circuit as per ohms law: resistance = voltage / current. A simple algebra formula gives the expected resistance in a circuit, say 9 ohms. Using a multimeter we carefully take the measurement and find 8.89 ohms. Build the same circuit a few more times and measure their resistances. Now maybe you find 9.20 and 8.922 and so on. This spread of measurements arises from the details of the actual, specific circuit being tested that are abstracted away in the simplicity of ohms law. The purity of the metal and the quality of the components are just two of the details that might be relevant in any particular case, there are thousands upon thousands of others.

With an actual measurement we encounter reality in all its uniqueness where more details, more evidence is included by the nature of the circumstances. Measurement is the bridge between theory and observation. It is writing a reality check. The data gathered will either conform to the expected result, increasing our confidence in the theoretical model or it will not. Given these spreads of observational data the question of just how close the value of the data needs to be to that predicted by theory and still be considered confirmation becomes critical. And it is just here that a funny thing happened on the way to the circus…

Turns out when you take a set of independent observations like this they disperse in that familiar pattern, the Bell Curve:

BellCurveDark blue is less than one standard deviation from the mean. For the normal distribution, this accounts for about 68% of the set, while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%. (From http://en.wikipedia.org/wiki/Standard_deviation)

 

Regular readers will recognize the shape from last week. By including more and more of the evidence a spread of sorts arises. We are trying on systems thinking by including more and more of the relevant detail, training to sense the shape of questions and answers as they appear to us in the real world.

So what is a probability? First let’s get an intuitive grasp of the concept. The prolific, gentlemanly “Prince of Mathematicians” (Bell  1937) Carl Friedrich Gauss at one time concerned himself with the errors that accompany astronomical observations. He published a few comments that Laplace immediately recognized the importance of. Laplace developed them and laid the foundation for modern probability theory. An astronomer records the latitude and longitude of a star’s location. Each observation differs in each direction from previous observations by some amount. How then should we consider this situation? For centuries the concern was that the errors of each of the observations would multiply. Astronomers such as Tycho Brahe had been averaging the observations for centuries. They seemed to have discovered by empirical means that instead of multiplying out of control the errors seemed to cancel out. It was Gauss who gave the mathematical proof that this is indeed the case. In a small comment he derived what we today call the Gaussian – Laplace curve. Most everyone is familiar with this figure; it is the normal or bell curve ubiquitous throughout statistics. The families of such curves are referred to as probability density functions.

Instead of saying that the star is really at the mid-point this curve describes the spread of uncertainty inherent in the collection of observations. The actual position can be anywhere within the scope of the curve though each position entails differing degrees of probability. Here a probability is a measure of the uncertainty both of our measurements and our understanding of causes. Other times a probability might be measuring an objective characteristic of the external world as, for example when measuring radioactive decay. Probability as a distribution was an amazing insight that was to play a fundamental role in the evolution of modern quantum mechanics where probability waves are used as a model of atomic structure.

Concern with the size of errors in collected data is the field of sampling theory and its significance tests. The correct hypothesis is known – the position of the star as determined by many previous observations and my star chart. The question concerns the data. Are the observations I record with my new telescopic alignment indicating it is properly calibrated? This is the type of question that concerned the creators of probability theory in the 18th and 19th centuries. They wanted to capture what could be said about the data to be expected when randomly drawing from a sample population. This is familiar to anyone who has taken a course in statistics. Every course introduces the ubiquitous, if morbid, Urn; an Urn contains 50 white balls and 20 red, what are the chances of drawing at least one red ball if 5 balls are drawn from the Urn and not replaced? The hypothesis is known, the contents of the Urn, and what we want to know is the distribution of the evidence we can expect from sampling it.

Sampling is the only means available to investigate the enormous complexity of the biosphere. The richness of the specifically existing actual objects and relationships exceeds our grasp any other way. But the roots of probability run even deeper than that. Many of the neurophysiological processing algorithms our senses use seem to rely on probability as well. It is not just the measuring but that which is measuring too, both are intimately and inescapably entwined with probabilities. Perhaps the most well-known is how the human eye has a blind spot where the optic nerve passes through the eyeball yet we do not see a black spot, void of anything. Instead the networks of neurons involved in processing optic signals interpolates what it expects would be in the external environment if it could see in this spot and fills the spot in with pure imagination. The brain performs a fundamentally probabilistic operation, guessing what is most probably there where it cannot actually see. An Amazon reviewer of Vision and Brain: How We Perceive the World put it well when they wrote, “human vision is a highly efficient guessing machine.” Indeed some researchers find that the roots of probability run even deeper than our sensory processing all the way down into how our brains do what they do. Bayesian Brains: Probabilistic Approaches to Neural Coding provides an approachable overview for those interested in taking a deeper look.

It should be obvious why these matters are important to the concerns of this blog. The majority of the evidence about the ecological crises presents itself to us in terms of probability. The IPCC report on climate change includes detailed treatment of the terms it uses for dealing with uncertainty. It is worth a substantial quote:

“Three different approaches are used to describe uncertainties each with a distinct form of language. Choices among and within these three approaches depend on both the nature of the information available and the authors’ expert judgment of the correctness and completeness of current scientific understanding.

Where uncertainty is assessed qualitatively, it is characterised by providing a relative sense of the amount and quality of evidence (that is, information from theory, observations or models indicating whether a belief or proposition is true or valid) and the degree of agreement (that is, the level of concurrence in the literature on a particular finding). This approach is used by WG III through a series of self-explanatory terms such as: high agreement, much evidence; high agreement, medium evidence; medium agreement, medium evidence; etc.

Where uncertainty is assessed more quantitatively using expert judgement of the correctness of underlying data, models or analyses, then the following scale of confidence levels is used to express the assessed chance of a finding being correct: very high confidence at least 9 out of 10; high confidence about 8 out of 10; medium confidence about 5 out of 10; low confidence about 2 out of 10; and very low confidence less than 1 out of 10.

Where uncertainty in specific outcomes is assessed using expert judgment and statistical analysis of a body of evidence (e.g. observations or model results), then the following likelihood ranges are used to express the assessed probability of occurrence: virtually certain >99%; extremely likely >95%; very likely >90%; likely >66%; more likely than not > 50%; about as likely as not 33% to 66%; unlikely <33%; very unlikely <10%; extremely unlikely <5%; exceptionally unlikely <1%.”

We saw in an earlier post how calculus provided science with a useful set of tools for creating mathematical models of events in a world of constant change. Probability provides an equally critical foundation for modern science with methods that are needed for logically interpreting the meaning of data gathered. Through the use of the rigor only mathematics can provide a consensus has been reached for these numerical operations that are no less objective in principal than the one we found with the ruler measuring the 11 inch stick.