Political arguments, custard elephants and the law of iterated expectations

The law of iterated expectations is a statistical fact which looks like this:

E[X] = E[ E[X | Y] ]

(Don't run away. The point is coming.)

In English, this says that the expected value of some variable X is equal to the expectation of its conditional expectations, given different values of another variable Y. (Don't run!)

Let's make it a bit more concrete. Suppose Y is a statement which may be true or false. Then:

E[X] = Probability(Y is true) * E[X, given Y is true]   +

            Probability(Y is false) * E[X, given Y is false]

The point to focus on is that if E[X, given Y is true] is higher than E[X], then E[X, given Y is false] cannot also be higher than X. You can show this with a bit of algebra, using that the probability of Y is strictly between 0 and 1. (Don't run! I swear that is the last of the algebra.)

For example, the expected value of a fair die throw is 3 and a half, and this is equal to

1/2 * 3 + 1/2 * 4

which is the expected value if the die lands on an odd number, times the chance of that; plus the expected value if the die comes up even, times the chance of that. You can cut the die roll up many different ways; the formula will always give you the right answer. Notice that 3.5 is strictly between 3 and 4.

Now, let's begin.

Here is an interesting sentence from Martin Jacques' excellent book When China Rules The World.

India's [post-colonial economic] performance was transformed, as the figures cited earlier for its economic growth illustrate, but Africa was left debilitated by the experience of the slave trade and then colonialism.

Jacques is referring to the fact that after the end of colonialism, GDP growth in Africa collapses – and even GDP levels, in some cases. As the context makes clear, the first part of this sentence claims:

1. India's GDP growth rose after independence, and this shows how bad colonialism was.

while the second part claims:

2. Africa's GDP growth fell after independence, and this shows how bad colonialism was.

Claims 1 and 2 violate the law of iterated expectations. Put Y as "a given country's GDP grows faster after independence from colonialism" and X as "how harmful colonialism was". If Y being true is evidence for the bad effects of colonialism, then Y being false cannot also be evidence for it. In fact, it must be evidence against it.

Someone might say: "You are oversimplifying absurdly. Of course there are complex links from colonialism to Africa's bad economic performance after independence. And of course India has its own, very different, history, which also features colonialism." This is true, but it doesn't affect the point. If your theory of history uses both Y and not-Y as evidence for the same overall claim, then your theory has a problem – not in the historical facts, but in how they are being interpreted.

I suspect that a lot of political arguments secretly violate the law of iterated expectations. The above was a particularly clear example in one sentence. More often, one person sees some evidence and goes "that proves my point"; but if they had seen the opposite, they would have said the same thing.

Or sometimes different people, on the same side of the argument, use contradictory pieces of evidence in different contexts. So, typically, scholars of African history will make the African argument above, and those of India will make the Indian argument. Either one may be right, but they cannot both be.

Here are some other examples. They are hypothetical, but they may sound familiar.

• An ethnic minority underperforms for a certain educational qualification. This shows that the relevant education institution is structurally racist. Students from the minority outperform for a different educational qualification. This alo shows that the education institution is structurally racist: it is setting the admission bar too high for those students.
• The government borrows in response to an economic shock. You think this is risky, and you predict rising inflation. But when inflation doesn't rise, that doesn't change your view. You just keep predicting it will rise in future.

Why political arguments, especially? For the usual reason: in politics and other morally-charged domains, we decide on our views and then find the evidence to support them.

Oh, and the custard elephants.

Why do elephants paint their feet yellow?
So they can hide upside down in custard.
Have you ever seen an elephant upside down in custard?
No? Shows what a good disguise it is, then.