B is for Bayesian Inference

Just  a note to say that I guest blogged for the University of Oxford's Mathematics Alphabet this week, on B is for Bayesian Inference.  Do check it out!

Ebola hunting

Here's a quick prediction for what happens now that the UK has started screening people for Ebola at Heathrow (soon to be extended to other airports and terminals).  They'll find a lot of people with minor stomach bugs of some sort.

Image: CDC

What price a preposition?

The headline to this article made me aware of a pretty big ambiguity in its contents: there's a big difference between saying that "in 30 seats UKIP are likely to win", and "UKIP are likely to win 30 seats". 

To see this, consider the statement: "Chelsea are likely to win every game they play this season".  Interpreted one way, you're just saying that Chelsea are favourites in each game they play (plausible).  Another meaning would be that Chelsea are likely to win all the games they play, which is exceptionally unlikely (no-one has ever come close).

Of course the results in different seats in general elections are strongly correlated, much more so than football matches.  But I'd still like to know what exactly the Guardian means.

Can a gamble ever be right or wrong?

A couple of weeks back I was feeling fairly smug, having put a bit of money on a 'No' vote in the Scottish referendum.  I then placed a few quid on UKIP to win the Heywood and Middleton by-election last Thursday, which they didn't.  But the latter result was quite close, so I felt that somehow it had been a 'good bet' to have made.  So what makes a person right or wrong to have placed a bet in the first place?  (Mathematically that is, leave your morals at your home page.)

Frog kissing. Or, why do we never learn?

A friend pointed me to this story about the average number of men (or rather frogs) a woman has to date before finding their 'Prince Charming' (apparently it's 15).  Let's leave aside for now how they came up with the number: I've no doubt the methodology conforms to the most rigorous standards that we've all come to expect from the internet.  I thought that this paragraph might give some people hope:

If you've made out with 11 men, great news! A new survey suggests you're only 4 away from Prince Charming. On average, anyway.

It's now my duty to crush that hope.

Distinguishing princes and frogs can be
difficult at first glance [Photo: Allen Warren]

The bad news is that (even if the rest of it is correct) the survey absolutely doesn't say that.  If you've been in 11 disastrous (but possibly ultimately affirming) relationships, you've guaranteed that the total number you end up in is at least 11.  But it could have been fewer than that (it happens apparently), so the possibility that you stayed with your childhood sweetheart is included in the average which arrives at 15.  Once you exclude that possibility, the average will only increase.

Just Instrument It

(With apologies to macro-economists.)

The amount of time you spend in education predicts your earnings quite strongly, and it's generally agreed (Simon Cowell aside) that if you want to do well in life, staying in education for longer is a good idea.

But how much effect does it have?  We could look at a survey of people's incomes and group them by education level, but this doesn't give a causal effect.  It might tell us that people who have a masters degree earn more during their lifetime, on average, than those who don't.  This could be because people from wealthy backgrounds can afford the tuition for a masters degree, and also have pals in the city who can help them get a big salary afterwards.  Or perhaps people who do masters degrees work harder than the rest.  We can't easily tell the difference: this problem is called confounding.

We can't tell whether time spent in education level causes earnings
to increase, or there's a third factor which affects both.

But is it causal? Defining causality

As I alluded to in my last post, defining what it means for $X$ to cause $Y$ is no simple task.  It is not an idea that can be defined in purely probabilistic terms, because it says something about the mechanisms underlying the system we are studying, and what will happen if we interfere with that system in some way.

Consider the example given at the end of the last post.  The headline was:

How a short nap can raise the risk of diabetes

The implication of this is that the risk of diabetes increases because of the nap.  But what does this mean?