Bayesian Death Match

 How likely are you to die of, say, covid versus, say, heart attack next year? If you look at official figures you are met with a variety of bewildering metrics. Furthermore it can be confusing to interpret what they mean in the first place. Here is a website that uses CDC figures to tell you what your odds are of dying of various causes, looking kind of like this:

Cause of Death	Odds of Dying
Heart disease	1 in 6
Cancer	1 in 7
Suicide	1 in 88
Fall	1 in 106

Of course, this doesn't mean you have a 16% chance of dying of a heart attack next year; it means when all is said and done, given that you died, the chance it was by heart attack was 16%. This is beginning to sound pretty Bayesian, so let's see if we can turn it into something more intuitive.

Manipulating probabilities by Bayes' Rule is powerful but often less than straightforward. Luckily there is a way to do it that is quick and easy. The trick is to think in terms of the logarithm of the odds ratio. You are used to thinking of probabilities as a number between 0 and 1; just think of them this way instead:

• -30 -- a billion to one against; your chance of winning a rigged lottery
• -20 -- a million to one against; your chance of being struck by lightning in a year
• -10 -- a thousand to one against;  your chance of flipping ten heads in a row
• 0 -- even odds
• 10 -- a thousand to one for; the chance you won't flip ten heads in a row 

and so forth. What we have done is taken the base-2 log of the odds ratio. Why would we do this?

The reason for using this log-odds form is that we can apply Bayes' Rule simply by adding them. Here's an example: the logodds for some given random average American to die (of any cause) next year is about -6 (roughly 1%). The logodds of dying in a car accident given that you died is -6.7. So the logodds of dying in a car crash next year is -12.7.

But we can do better than that. You aren't a random American: you can improve your estimate of the prior by knowing, for example, your age. CDC says:

This translates to 

Age	Logodds
20	-10.5
30	-9.6
40	-9
50	-8
60	-6.8
70	-5.8
80	-4.5
90	-2.9

So rather than start with just -6, you'd start with the number associated with your age. Or you could have a table by sex, or whatever other division you thought made a difference.

Then add the number for the thing you're worried about. Dying from a fall is -6.7. So if you're 20, your total risk from falls is -17.2; you'll outlive Methuselah. But if you're 90, it's -9.6, well within the range of things to worry about.

The numbers for heart disease, cancer, and covid all stand pretty close to -2.6. Do the math.