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.

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