Subjectivity

Subjectivity is the Achilles’ Heel of modern science.

The discovery of quantum mechanics can be instructive. At the dawn of the twentieth century physicists were rather confident their discipline had captured the most essential aspects of how the physical world works. There were just a few experiments and observations that did not fit the prevailing theories, but for the most part the work of physics in the twentieth century would be one of, as Albert Michelson quipped (and lived to regret!), filling in the sixth decimal place. By 1927 the whole apple cart of our physical understanding would be overturned.

The items that did not fit the standard models of the physics of the day seemed to be rather small discrepancies. Three experimental results in particular were troubling. The first was what was known as the ultraviolet catastrophe associated with black body radiation, the solution of which would lead to Plank’s quantum of energy replacing the previous conception of energy as continuous. The second experiment was the photoelectric effect, which Einstein’s explained provided evidence that light could act as particles as well as waves. The third was bright line optical spectra, which would lead to the Bohr atom as the first atomic model to account for the discrete energy states being observed.

It is important to understand just how successful Newton’s gravitational theory had been in classical physics. Objects in motion were subject to rigorous analysis with the tool of the calculus and the conceptual abstractions of force and momentum with such accuracy we still use the same techniques in our age of satellites. It is also important to understand just how successful Maxwell’s wave theory of electromagnetism had been in classical physics to appreciate just how radical the coming of quantum mechanics really was. It was not just that the universe once thought to be continuous became discontinuous and radically momentary. A deterministic universe gave way to one ruled by probability.

Philosophy was there before science. Kant had identified space and time as absolute categories of thought, presenting us with the picture of the mindless, clockwork universe as the scaffolding on which the very ability to think at all depended. When relativity removed the absolute nature from time and space, the way was open for Schopenhauer to explain the world as will and representation. The subjectivity, the will, of what had been discovered in our hunt for objectivity was laid bare. The mechanical universe of classical physics, the one made in the image of our machines, gave way to, well, no one is quite sure just yet what the new picture of reality is trying to teach us. There are, however, clues.

The difference between determinism and probability is a very big deal. To glance for a moment at the headlines: the fundamentalist fanaticism of the true believer is built brick by brick from their certainties. Those who hold their truths more humbly, recognizing the limitations of human understanding, are less likely to forget logical inferences are founded on probability.

Classical science was understood to be dedicated to seeking a type of truth that was completely objective. The revolutionary scientific method insisted that opinions no longer be taken as facts. We learned to insist that if you make a claim about what is actually real and what is not, there needed to be evidence to back it up. The mathematical methods the sciences use are all designed to provide the type of knowledge that relies on measurable evidence. It was a revolution in where the ultimate authority, the final court of appeal, was to be found. No longer could the king, saint or pope declare what was and what was not, simply by virtue of their position. Facts took on a new importance. When the scientific revolution began this was indeed a very revolutionary position to take. It was also democratic. These scientific measurements could be taken by anyone anywhere and each person could prove for themselves the experiment properly performed lead to consistent results. The acceleration of gravity, as we learned in school, is 9.8 meters per second per second at sea level. It is so as much for a Chinaman as it is for an Englishman.

For those who really understood what this was all about the authority did not move from the kings, saints, and popes to the scientific experts. The authority moved into each person’s own eyes and hands by which they could handle the evidence for themselves and, most importantly, the authority of each person’s reason became recognized as the final court of appeal. Power can torture a man and make him recant his beliefs but only what is undeniably true for his reason carries the real power to persuade. (Mindful Ecology has suggested since its inception that every home that can should have mind tools at the ready; a telescope, microscope and access to encyclopedias. It is not what you read or watch that teaches best, it is what you do.)

Science insists its investigations remain grounded in the realms of evidence, which works to keep it deeply embodied in reality. This was a powerful blow against superstition. It was a liberation from our inherent gullibility and the conmen that have ever been at the ready to exploit it. On the other hand, the pursuit of scientific theory involves finding the right abstraction, the one that will capture the essence of what the embodied evidence is indicating. We do not do good science when we have one law of gravity for apples and another for planets. Those mathematical abstractions exist in a realm where the body of the thinker no longer seems to be playing any vital role at all. In the Platonic realm of pure mathematics where is blood and flesh? Over time the abstract was given more respect than the particular, standing things on their head. Our societies became even more committed to Descartes Error: reason defined as thought wholly uninfluenced by emotion came to be considered the summit of humanity’s capacity for understanding.

With the coming of quantum mechanics and relativity the role of the observer could no longer be ignored. Subjectivity is the blackbody radiation of our times, an indication that something fundamental is missing from our view of what is really going on: we do not know what the role of consciousness is in the universe.

It Has Begun

“I don’t know about you but I feel like Jung was certainly right. I mean, I have mentioned before that what we do affects the next seven generations. In other words, I am carrying the history and experiences of the past seven generations. Some will say that it is not fair, that it means I am carrying baggage. But you have to remember that it goes both ways. I am also carrying the beauty, strength and knowledge of the past seven generations. Sometimes it is also about remembering or investigating where we come from, so we know better who we are today.

The history of the Native Americans is sadly filled with trauma and what I would qualify as genocides. Massacres such as Wounded Knee, need to be remembered as they affect the soul of all. Such massacres also affect the land they took place upon. The memory of what happened lives within the Earth. The bodies, the blood lives within the soil. If you think about how Native American culture emphasizes the connection to the Earth, a wound to the Earth is a wound to the people. It is a wound to the earth-connected side we all have, thus a wound to the soul. The feelings and the hurt of those who passed away on the battlefields do not die with them. They remain in each of us. The Land holds our stories, the land will evoke our personal and collective stories, it will remind us of them. As the land is also living. Violence to the people or the land led to the suffering of the following generations, as it is stored in our collective unconscious or psyche.”
Carl Jung’s Collective Unconscious and Native Americans

“No law shall be passed that harms the children.”
Native American Tribal Counsel

 

It is all together too easy to lose our personal power. One person gazing with horror at the non-stop carbon dioxide production of our global footprint – what are we to do? Words by the billions have already been written, detailing what we know about humanity’s ecological relationships with the biosphere. Studies have been funded and conferences have been held, speeches have been given and protests have been organized. Prayers and songs have been offered, tears and blood have been shed.

Still the amount of carbon dioxide pollution increases at a frightening rate year after year. Still each year is a record breaking one, warmer than the last. We are all living in a slow motion train wreck. Those with window seats are traumatized, and in my mind, they are the lucky ones. Those who are called to become mindful of ecological relationships between the human footprint and the biosphere are given sacred knowledge. We cannot use it, we can allow it to use us.

If we ask how we can make a difference that will really make a difference, it is difficult to imagine anything we could do that has not already been done. There is not, in fact, much a single individual can do to change the trajectories we are on. It does no good to pretend otherwise. This is the shock of the horrified, the shock that comes to those who are given a glimpse of the Juggernaut we have built. Have you seen it? Have you watched Homo Colossus tear up the earth? It’s metallic maw chews up rain forests and ocean reefs then spits out cancerous waters darkened with sickness and starvation. If we are honest with ourselves, it is as if some part of ourselves simply flat-lines staring at the wall of ignorant indifference. We are numbed by the planetary powers in play of truly titanic proportions.

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Yesterday I saw a graffiti of ‘Trump’ across an ecology sign along a park walkway. Gave me the feeling of darkening Nazi skies. There is a perception that ecologists are freaks; forever going on about chipmunks and creeks, moss killer and robins. Freaks that should just be run over if they insist on standing in the way of making America Great Again.

It doesn’t do any good to pretend we are not who and what we are. We are tempted to pretend we do not have a point of view running as deep as these boulders. As if by denying these passionate, simple commitments made in our hearts we could assure the world that we too are just like everyone else. Something to bear in mind when thinking about these things. Your own simplicity is the path.

It was an interesting victory a few days ago for the long patient Standing Rock protest. Their prayer and ceremony was powerfully effective on a number of levels. The ripples from these events are bound to play out along lines both joyful and sorrowful, beautiful and ugly for a long time to come. People mindful of ecology will most certainly want to keep an eye on this pipeline project as it has become a rumbling of earth spirituality through the federal government of the United States. If my intuition is right there is considerably more riding on the wings of those prayer feathers than first meets the eye. Water protectors are of the elemental realm. They have witnessed to a level of purity and fundamental truth about our times. The troubled American psyche marks the moment: water turned against water protectors, veterans involved, treaty rights of this land’s indigenous peoples questioned again, the Army Corp of Engineers involved, energy corporations involved, ecological legal regulations involved, and all attended by a popular uprising of people insisting that the wholesale destruction of the earth must stop. Now.

Mark this moment. Grandfathers, please pray for us.

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Sustainability (def.): of, relating to, or being a method of harvesting or using a resource so that the resource is not depleted or permanently damaged.

The changes to the ocean acidification recorded in the last hundred years will take ten-thousand years to work their way through the system. Are there words for this? Has a generation ever before claimed so much for themselves alone?

This is a simple boulder of truth: that which cannot be sustained, will not be sustained.

I have no more words.

Let’s just sit and watch the reason we brought this disaster upon our children, the wonderful bounty of our happy modern lives. Koyaanisqatsi.

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.