Sep 262014

As they fire their what happens if questions, small children regard the answer as perfect evidence, just as they have been doing in infancy on their own through repeated experiment. As a baby drops a ball from his hands, he learns that the ball always lands on the floor, and that it never lands on the floor unless he drops it.

Perfect evidence is if and only if, necessary and sufficient. Likewise, as a child asks his dad what happens if he puts a finger on a flame, he intends the answer to mean that the finger will always burn if he puts it there, and that it won’t unless he does.

Dad’s explanation of why this is so strengthens the child’s belief by assigning a cause to the effect. But such fatal move marks the end of parents’ aura of infallibility. Once the why-chain monster is unleashed, the child will soon realise that mum and dad do not have all the answers. Each explanation begets a new question and there is no super-ring at the end of the chain – a brute conundrum which he will learn to deal with one way or another, but will never be able to solve.

As they give up on what soon turns out to be a weary why-game, children learn to accept and get on with local explanations. Parents continue to be their main source of evidence. Some of it will still be perfect and some will be conclusive. But an increasing proportion will be imperfect and inconclusive.

For example: a child picks up a hardened chewing gum from the floor and asks: ‘Can I put it in my mouth?’ After a horrified ‘No!’ comes the next question: ‘Why? What happens if I do?’, to which the correct answer: ‘Nothing, most likely’, is clearly inadequate. The right answer is some variation of: ‘It is dangerous’, which opens up a whole new world: the world of possibilities, where things can happen, rather than will happen. ‘Dangerous’ is an aptly concise way of saying: Take 20,000 children, have half of them chew a gum picked from the floor and the other half a gum from a sealed packet. After a while, some children will get sick. You will see that there will be more sick children among those who chewed gums picked from the floor than among those who chewed packed gums. In a nightmare, the child could reply: ‘Really? Has such an experiment actually been performed?’ Thankfully, it doesn’t happen: children trust their parents.

Trust is measured by the Likelihood Ratio. In this case, the tested hypothesis is: ‘Chewing the floor gum is bad for me’, and the evidence is: ‘Dad says so’. The Likelihood Ratio is the ratio between TPR: the probability that dad says that chewing the floor gum is bad, given that it really is bad, and FPR: the probability that dad says it is bad, given that it actually isn’t. An infallible dad – the perfect hero of small children – has TPR=1 and FPR=0: he is never wrong. Children soon realise that is not the case – a developmental stage that smart parents accompany and encourage and dumb parents vainly oppose. Most parental evidence is imperfect. Still, while no longer infinite, parents’ Likelihood Ratios remain large and, multiplied by prior odds – equal to one for most hypotheses to perfectly ignorant children – determine children’s posterior odds: if dad says so, it must be right – well, almost certainly.

It is, alas, a temporary biological advantage. As children grow up, their trust into whatever parents say is bound to be challenged by other sources of soft evidence – other relatives, teachers, friends, and then TV, books, internet and the whole wide world. ‘Dad is right’ becomes less and less a forgone conclusion. As they add new sources of evidence, children learn to multiply their Likelihood Ratios, in the same way that, since the dawn of civilisation, the Law has been using the judgement of reputable people to reach verdicts. Parents will no longer be the only jurors and, in most cases, will not even be part of the jury. Which, of course, is as it should be.

The only way parental trust can produce a lasting influence on children is by permeating their priors. It is what we call, in its broadest sense, education – a set of beliefs, values, principles and priorities that form the basis upon which evidence is evaluated. Education is not just teaching what happens if, what is true or false, right or wrong: it is explaining why.

Since childhood, we test hypotheses using available evidence to update our priors. Whether we judge a hypothesis to be true or false depends on the evidence, but is based on priors. Evidence is placed within the confines of what we already know. Strong priors, founded on good explanations, help us avoid prior indifference.

This is the power of Why. We have seen it in Tversky and Kahneman’s cab problem. When people are just told that 85% of cabs are Green, they go along with the witness. But when they are told that Green cabs are involved in 85% of the accidents, they successfully reduce their posterior probabilities close to the correct value. They do so because they have a good reason to believe that Blue cabs are less likely to be involved in an accident: sloppier Green cab drivers. Similarly, we have seen it in Newcomb’s Paradox. If we are just told that Dr Wise is infallible, we are tempted to open both boxes. But if, as in the Janken version, we are told why the robot is infallible, we easily recognize how foolish it would be to bet against it.

Likewise, in our child footballer story it is easy to imagine that, if the father had a good reason to be sceptical about his child’s chances of success, he would have taken the coach’s opinion with a large grain of salt. Having a good reason to doubt homeopathic medicine, supernatural powers, conspiracy theories and assorted nonsense provides an effective shield against seemingly compelling evidence. If Iago had not been able to melt Othello’s solid priors on his spouse’s loyalty, the Moor would not have killed Desdemona. In general, when evidence runs counter to well-founded priors, updates occur, by and large, according to Bayes’ Theorem.

The bad news, however, is that Bayes’ Theorem works in the same way on equally strong but ill-founded priors. Homeopaths, spiritualists and conspiracists will not be swayed by the most genuinely compelling evidence. Like any power, Why has a dark side. A good explanation is a story that satisfies us and stops us asking more questions. We are satisfied that the earth is round. The trouble is that people can find satisfaction in very odd places.

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