On 1 Jan 2018, I made the following entry into my journal
- Will Trump still be president? Yes. (P = 80%)
- Will Mueller team link Russia to Trump: a) To Trump campaign yes (P = 60%); b) to Trump No (P = 70%)
- Will Crypto continue to rise? Yes. (P = 60%)
- Will the stock market end its rise? No. (P = 55%)
- Will Republicans lose control of the house in November? Yes. (P = 75%)
- Will there be a war with North Korea? No. (P = 95%)
- Will the New York Times go out of business? No. (P = 85%)
- Will we cure one specific type of cancer? Yes. (P = 60%)
- Will there be at least one Bayesian-based company that will raise Series B? (P = 70%)
I also said that I would compute my gain/loss using a hypothetical payoff function: \(100*\text{log}(2p) \) if I am right and \(100*\text{log}(2 * (1-p)) \) if I am wrong, where p is the probability I assign to the event occurring. We could use any base for a log but base 2 is natural as it compensates at the notional value ($100) if the bet is made with probability 1. I will describe why this particular payoff function makes sense in another post. (The tacit assumption here is that I would have been able to find a counterparty for each one of these bets, which is debatable.)
- Trump is still president: \(100*\text{log2}(2*0.80) = 68\)
- Mueller linked Trump campaign to Russia. The word link was not defined. I think it is reasonable to assume that the link had been established, but I could see how if my counterparty was a strong Trump supported, they could dispute this claim. Anyway: \(100*\text{log2}(2*0.60) = 26\)
- Mueller linked Trump to Russia. Same as above in terms of the likelihood of it being contested, but think I lost this bet: \(100*\text{log2}(2*0.30) = -74\)
- Crypto did not continue to rise: \(100*\text{log2}(2*0.40) = -32\)
- Stock market ended its rise: \(100*\text{log2}(2*0.45) = -15\)
- Republicans lost control of the house in November: \(100*\text{log2}(2*0.75) = 58\)
- Thankfully, there is no war with North Korea: \(100*\text{log2}(2*0.95) = 93\)
- New York Times is still in business: \(100*\text{log2}(2*0.85) = 76\)
- I am not sure what made me so
optimimistic regarding the cure for one type of cancer. Currently, the most promising cancer therapied are PD-1/PD-L1 immune checkpoint inhibitors and there have been documented cases for people who become cancer-free after being treated with one of these drugs, but I think it would be too generous to say that we have cured one type of cancer. Perhaps more impressively, Luxturna will cure your blindness with one shot to each eye if a) you have a rare form of blindness that this drug targets and b) you have $850,000 to spend. \(100*\text{log2}(2*0.40) = -32\) - There were a few startups based on the Bayesian paradigm and Gamalon came close with a $20M Series A round, but none raised Series B to my knowledge: \(100*\text{log2}(2*0.30) = -74\)
To summarize, I am up $94. Is this good or bad? It depends. A good forecaster is well-calibrated and we do not enough here to compute my calibration. The second condition is that for the same level of calibration we prefer a forecaster that predicts with higher certainty, a concept known as sharpness. Check out this paper if you are curious.