Your definition of the Heaviside step function has H(0) = 1.
Your definition of L has L(0) = 1/2, so you're not really taking the derivative of the same function.
I don't really believe nonstandard analysis helps us differentiate the Heaviside step function. You have found a function that is quite a lot like the step function and shown that it has a derivative (maybe), but I would need to be convinced that all functions have the same derivative to be convinced that something meaningful is going on. (And since all your derivatives have different values, this seems like a not useful definition of a derivative)
The log-returns are not linear in the bits. (They aren't even constant for a given level of bits.)
For example, say the market is 1:1, and you have 1 bit of information: you think the odds are 1:2, then Kelly betting you will bet 1/3 of your bankroll and expect to make a ~20% log-return.
Say the market was 1:2, and you had 1 bit of information: you think the odds are 1:4, then Kelly betting, you will bet 1/5 of your bankroll and expect to make a ~27% log-return.
We've already determined that quite different returns can be obtained for the same amount of infor...
When this paradox gets talked about, people rarely bring up the caveat that to make the math nice you're supposed to keep rejecting this first bet over a potentially broad range of wealth.
This is exactly the first thing I bring up when people talk about this.
...But counter-caveat: you don't actually need a range of $1,000,000,000. Betting $1000 against $5000, or $1000 against $10,000, still sounds appealing, but the benefit of the winnings is squished against the ceiling of seven hundred and sixty nine utilons all the same. The logic doesn't require that
Thus, an attacker, knowing this, could only reasonably expect to demand half the amount to get paid.
Who bears the cost of a tax depends on the elasticities of supply and demand. In the case of a ransomware attack, I would expect the vast majority of the burden to fall on the victim.
I wrote about exactly this recently- https://www.lesswrong.com/posts/zLnHk9udC28D34GBB/prediction-markets-aren-t-magic
I don't give much weight to his diagnosis of problematic group decision mechanisms
I have quite a lot of time for it personally.
The world is dominated by a lot of large organizations that have a lot of dysfunction. Anybody over the age of 40 will just agree with me on this. I think it's pretty hard to find anybody who would disagree about that who's been around the world. Our world is full of big organizations that just make a lot of bad decisions because they find it hard to aggregate information from all the different people.
This is roughly Hanson'...
So the first question is: "how much should we expect the sample mean to move?".
If the current state is , and we see a sample of (where is going to be 0 or 1 based on whether or not we have heads or tails), then the expected change is:
In these steps we are using the facts that ( is independent of the previous samples, and the distribution of is Bernoulli with . (So and ...
Whoops. Good catch. Fixing
x is the result of the (n+1)th draw sigma is the standard deviation after the first n draws pnl is the profit and loss the bettor can expect to earn
Prediction markets generate information. Information is valuable as a public good. Failure of public good provision is not a failure of prediction markets.
I think you've slightly missed my point. My claim is narrower than this. I'm saying that prediction markets have a concrete issue which means you should expect them to be less efficient at gathering data than alternatives. Even if information is a public good, it might not be worth as much as prediction markets would charge to find that information. Imagine if the cost of information via a prediction market was exponential in the cost of information gathering, that wouldn't mean the right answer is to subsidise prediction markets more.
If you have another suggestion for a title, I'd be happy to use it
Even if there is no acceptable way to share the data semi-anonymously outside of match group, the arguments for prediction markets still apply within match group. A well designed prediction market would still be a better way to distribute internal resources and rewards amongst competing data science teams within match group.
I used to think things like this, but now I disagree, and actually think it's fairly unlikely this is the case.
Sure - but that answer doesn't explain their relative lack of success in other countries (eg the UK)
Additionally, where prediction markets work well (eg sports betting, political betting) there is a thriving offshore market catering to US customers.
This post triggered me a bit, so I ended up writing one of my own.
I agree the entire thing is about how to subsidise the markets, but I think you're overestimating how good markets are as a mechanism for subsidising forecasting (in general). Specifically for your examples:
I'm excited about the potential of conditional prediction markets to improve on them and solve two-sided adverse selection.
This applies to roughly the entire post, but I see an awful lot of magical thinking in this space. What is the actual mechanism by which you think prediction markets will solve these problems?
In order to get a good prediction from a market you need traders to put prices in the right places. This means you need to subsidise the markets. Whether or not a subsidised prediction market is going to be cheaper for the equivalent level of forecast than paying another 3rd party (as is currently the case in most of your examples) is very unclear to me
A thing Larry Summers once said that seems relevant, from Elizabeth Warren:
He said something very similar to Yanis Varoufakis (https://www.globaljustice.org.uk/blog/2017/06/when-yanis-met-prince-darkness-extract-adults-room/) and now I like to assume he goes around saying this to everyone
No, it's fairly straightforward to see this won't work
Let N be the random variable denoting the number of rounds. Let x = p*w+(1-p)*l where p is probability of winning and w=1-f+o*f, l=1-f the amounts we win or lose betting a fraction f of our wealth.
Then the value we care about is E[x^N], which is the moment generating function of X evaluated at log(x). Since our mgf is increasing as a function of x, we want to maximise x. ie our linear utility doesn't change
Yes? 1/ it's not in their mandate 2/ they've never done it before (I guess you could argue the UK did for in 2022, but I'm not sure this is quite the same) 3/ it's not clear that this form of QE would have the effect you're expecting on long end yields
I absolutely do not recommend shorting long-dated bonds. However, if I did want to do so a a retail investor, I would maintain a rolling short in CME treasury futures. Longest future is UB. You'd need to roll your short once every 3 months, and you'd also want to adjust the size each time, given that the changing CTD means that the same number of contracts doesn't necessarily mean the same amount of risk each expiry.
Err... just so I'm clear lots of money being printed will devalue those long dated bonds even more, making the bond short an even better trade? (Or are you talking about some kind of YCC scenario?)
average returns
I think the disagreement here is on what "average" means. All-in maximises the arithmetic average return. Kelly maximises the geometric average. Which average is more relevant is equivalent to the Kelly debate though, so hard to say much more
Wouldn’t You Prefer a Good Game of Chess?
I assume this was supposed to be a WarGames reference, in which case I think it should be a "nice" game of chess.
Yeah, and it doesn't adjust for taxes there either. I thought this was less of an issue when comparing rents to owning though, as the same error should affect both equally.
This doesn't seem to account for property taxes, which I expect would change the story quite a bit for the US.
I’d also add that female labor force participation rates will move these numbers around some. Their calculations assume all countries have 50% female participation when calculating income, when it actually varies from 11%-85% or so.
This seems needlessly narrow minded. Just because AI is better than humans doesn't make it uniformly better than humans in all subtasks of chess.
I don't know enough about the specifics that this guy is talking about (I am not an expert) but I do know that until the release of NN-based algorithms most top players were still comfortable talking about positions where the computer was mis-evaluating positions soon out of the opening.
To take another more concrete example - computers were much better than humans in 2004, and yet Peter Leko still managed to refute a computer prepared line OTB in a world championship game.
Agreed - as I said, the most important things are compute and dilligence. Just because a large fraction of the top games are draws doesn't really say much about whether or not there is an edge being given by the humans (A large fraction of elite chess games are draws, but no-one doubts there are differences in skill level there). Really you'd want to see Jon Edward's setup vs a completely untweaked engine being administered by a novice.
I believe the answer is potentially. The main things which matter in high-level correspondence chess are:
Although I don't think either of those are really relevant. The really relevant bit is (apparently) planning:
For me, the key is planning, which computers do not do well — Petrosian-like evaluations of where pieces belong, what exchanges are needed, and what move orders are most precise within the long-term plan.
(From this interview with Jon Edwards (reigning correspondence world champion) from...
Just in case anyone is struggling to find the relevant bits of the the codebase, my best guess is the link for the collections folder in github is now here.
You are looking in "views.ts" eg .../collections/comments/views.ts
The best thing to search for (I found) was ".addView(" and see what fits your requirements
I feel in all these contexts odds are better than log-odds.
Log-odds simplifies Bayesian calculations: so does odds. (The addition becomes multiplication)
Every number is meaningful: every positive number is meaningful and the numbers are clearer. I can tell you intuitively what 4:1 or 1:4 means. I can't tell you what -2.4 means quickly, especially if I have to keep specifying a base.
Certainty is infinite: same is true for odds
Negation is the complement and 0 is neutral: Inverse is the complement and 1 is neutral. 1:1 means "I don't know" and 1:x is the inverse of x:1. Both ot these are intuitive to me.
No - I think probability is the thing supposed to be a martingale, but I might be being dumb here.
So, what do you think? Does this method seem at all promising? I'm debating with myself whether I should begin using SPIES on Metaculus or elsewhere.
I'm not super impressed tbh. I don't see "give a 90% confidence interval for x" as a question which comes up frequently? (At least in the context of eliciting forecasts and estimates from humans - it comes up quite a bit in data analysis).
For example, I don't really understand how you'd use it as a method on Metaculus. Metaculus has 2 question types - binary and continuous. For binary you have to give the prob...
17. Unemployment below five percent in December: 73 (Kalshi said 92% that unemployment never goes above 6%; 49 from Manifold)
I'm not sure exactly how you're converting 92% unemployment < 6% to < 5%, but I'm not entirely convinced by your methodology?
15. The Fed ends up doing more than its currently forecast three interest rate hikes: None (couldn't find any markets)
Looking at the SOFR Dec-22 3M futures 99.25/99.125 put spread on the 14-Feb, I put this probability at ~84%.
Thanks for doing this, I started doing it before I saw your competition an...
And one way to accomplish that would be to bet on what percentage of bets are on "uncertainty" vs. a prediction.
How do you plan on incentivising people to bet on "uncertainty"? All the ways I can think of lead to people either gaming the index, or turning uncertainty into a KBC.
The market and most of the indicators you mentioned would be dominated by the 60 that placed large bets
I disagree with this. Volatility, liquidity, # predictors, spread of forecasts will all be affected by the fact that 20 people aren't willing to get involved. I'm not sure what information you think is being lost by people stepping away? (I guess the difference between "the market is wrong" and "the market is uninteresting"?)
There are a bunch of different metrics which you could look at on a prediction market / prediction platform to gauge how "uncertain" the forecast is:
Prediction markets function best when liquidity is high, but they break completely if the liquidity exceeds the price of influencing the outcome. Prediction markets function only in situations where outcomes are expensive to influence.
There are a ton of fun examples of this failing:
I don't know enough about how equities trade during earnings, but I do know a little about how some other products trade during data releases and while people are speaking.
In general, the vast, vast, vast majority of liquidity is withdrawn from the market before the release. There will be a few stale orders people have left by accident + a few orders left in at levels deemed ridiculously unlikely. As soon as the data is released, the fastest players will general send quotes making a (fairly wide market) around their estimate of the fair price. Over time (a...
I agree identifying model failure is something people can be good at (although I find people often forget to consider it). Pricing it they are usually pretty bad at.
I'd personally be more interested in asking someone for their 95% CI than their 68% CI, if I had to ask them for exactly one of the two. (Although it might again depend on what exactly I plain to do with this estimate.)
I'm usually much more interested in a 68% CI (or a 50% CI) than a 95% CI because:
Under what assumption?
1/ You aren't "[assuming] the errors are normally distributed". (Since a mixture of two normals isn't normal) in what you've written above.
2/ If your assumption is then yes, I agree the median of is ~0.45 (although
from scipy import stats
stats.chi2.ppf(.5, df=1)
>>> 0.454936
would have been an easier way to illustrate your point). I think this is actually the assumption you're making. [Which is a horrible assumption, because if it were true, you would already be perfectly calibrated].
3/ I guess ...
I think the controversy is mostly irrelevant at this point. Leela performed comparably to Stockfish in the latest TCEC season and is based on Alpha Zero. It has most of the "romantic" properties mentioned in the post.
That isn't a "simple" observation.
Consider an error which is 0.5 22% of the time, 1.1 78% of the time. The squared errors are 0.25 and 1.21. The median error is 1.1 > 1. (The mean squared error is 1)
Metaculus uses the cdf of the predicted distribution which is better If you have lots of predictions, my scheme gives an actionable number faster
You keep claiming this, but I don't understand why you think this
If you suck like me and get a prediction very close then I would probably say: that sometimes happen :) note I assume the average squared error should be 1, which means most errors are less than 1, because 02+22=2>1
I assume you're making some unspoken assumptions here, because is not enough to say that. A naive application of Chebyshev's inequality would just say that .
To be more concrete, if you were very weird, and either end up forecasting 0.5 s.d. or 1.1 s.d. away, (still with mean 0 and average...
Go to your profile page. (Will be something like https://www.metaculus.com/accounts/profile/{some number}/). Then in the track record section, switch from Brier Score to "Log Score (continuous)"
I'd be happy to.
The 2000-2021 VIX has averaged 19.7, sp500 annualized vol 18.1.
I think you're trying to say something here like 18.1 <= 19.7, therefore VIX (and by extension) options are expensive. This is an error. I explain more in detail here, but in short you're comparing expected variance and expected volatility which aren't the same thing.
...From a 2ndary source: "The mean of the realistic volatility risk premium since 2000 has been 11% of implied volatility, with a standard deviation of roughly 15%-points" from https://www.sr-sv.com/realistic-volatility-risk-premia
I still think you're missing my point.
If you're making ~20 predictions a year, you shouldn't be doing any funky math to analyse your forecasts. Just go through each one after the fact and decide whether or not the forecast was sensible with the benefit of hindsight.
I am even explaining what an normal distribution is because I do not expect my audience to know...
I think this is exactly my point, if someone doesn't know what a normal distribution is, maybe they should be looking at their forecasts in a fuzzier way than trying to back fit some model to them.
...A
The original post also addresses this suggestion