1

Suppose you have $n$ forecasts for an event which can have discrete outcomes, for example $X$, $Y$ and $Z$. Let forecast $i$ give the probabilty of each event occuring as $x_i$, $y_i$ and $z_i$ ($x_i+y_i+z_i=1$). What methods can be used to combine the individual forecasts into a 'joint' forecast in this situation? I.e. how can I get sensible probabilties $x=f(x_1,y_1,z_1,x_2,...)$, $y=g(x_1,y_1,z_1,x_2,...)$ and $z=h(x_1,y_1,z_1,x_2,...)$? ($x+y+z=1$)

If I were dealing with point forecasts for an event with real valued outcomes, I might have used an equal weighted sum of individual forecasts. With discrete outcomes, the situation is not so clear.

Comp_Warrior
  • 2,075
  • 1
  • 20
  • 35

1 Answers1

1

I am not sure of the answer, but what you describe sounds like working with compositional data. I found an interesting summary of techniques that might be helpful, and it appears that you might want to use a log-ratio transform.

The simplest version of a log-ration transform is to take one of your probabilities as the baseline, say $x$ and work with $\hat{y}_i=\log (y_i/x_i)$ and $\hat{z}_i=\log (z_i/x_i)$. I believe you could then calculate the mean of each ($\sum_i{\hat{y}_i}/n$ and $\sum_i{\hat{z}_i}/n$) and then back-transform to probabilities, calculating the baseline mean probability as 1 minus the sum of your back-transformed means for $y$ and $z$.

Again, I don't know if this is general practice with probabilities, but hopefully it at least gives you some new concepts and keywords to follow. (I believe that the log-ratio transform takes your probabilities out of the simplex to the reals so you can use your usual techniques.)

Wayne
  • 19,981
  • 4
  • 50
  • 99