## Thursday, July 26, 2012

### Shangri La Experiment: More Priors

Forgot that I should write down the predictions of each of my three theories in advance.
I did make some in my head, but I never got round to writing them down.

As I remember my prior predictions were:

There'll be two measurable variables:

Favourite Belt Notch (1,2,3,4,5,6) and Perceptible Loss of Appetite (yes/no)

I'll build prior probabilities by distributing 100 points around the likely predictions, and then putting 5 points in neighbouring cells, and 1s everywhere else.

Willpower (no loss of appetite, some increased girth)

1   2  3 4 5 6
yes  5   5  5 1 1 1
no   75 25  5 1 1 1

total 126

Shangri-La (huge loss of appetite, loss of girth)

1 2  3  4  5 6
yes 1 5 50 25 25 5
no  1 5  5  5  5 5

total 137

Helplessness (some loss of appetite, may or may not be noticeable, no change of girth)

1   2 3 4 5 6
yes 5  50 5 1 1 1
no  5  50 5 1 1 1

total 126

So for instance, if at the end of the month I am on belt notch 2, with noticeable loss of appetite, which was my actual prediction, then I can Bayesify matters thus:

prior H60:W39:S1

likelihood ratios H 50/126: W 5/126: S 5/137

new beliefs H 60*50/126: W 39*5/126: S 1*5/137

H 60*50*137: W 39*5*137: S 5*126

H 411000: W 26715: S 630

rounding:

H 937: W 60: S 1

Which looks right. If the experiment agrees with my pet theory, then I should believe in it lots.
If it disagrees with a mad theory that I don't believe in to start with, I should pretty much give up on that theory, and Willpower is getting dissed because it doesn't say I should notice a loss of appetite if I eat more.

Of course, loss of appetite is highly subjective, and I'm going to be tempted to try to make my experiment come out the way I want, which is essentially 'go, go Shangri-La, fuck off Willpower', so I should also consider what goes on if I report 'no loss of appetite, notch 2'

prior H60:W39:S1

likelihood ratios H 50/126:  W 25/126: S 5/137

new beliefs H 60*50*137: W 39*25*137: S 5*126

H 411000 : W 133575 : S 630

H753: W244: S1

Which also looks right. W is getting penalised for favouring weight gain that didn't happen more than H is getting penalised for hedging its bets on appetite loss. S is getting kicked out into the dark for getting it wrong on both counts.

etc. etc.