All of Jan Christian Refsgaard's Comments + Replies

This might help you https://github.com/MaksimIM/JaynesProbabilityTheory

But to be honest I did very few of the exercises, from chapter 4 and onward most of the stuff Jayne says are "over complicated" in the sense that he derives some fancy function, but that is actually just the poison likelihood or whatever, so as long as you can follow the math sufficiently to get a feel for what the text says, then you can enjoy that all of statistics is derivable from his axioms, but you don't have to be able to derive it yourself, and if you ever want to do actual Baye... (read more)

I am not aware of Savage much apart from both Bayesian and Frequentists not liking him. And I did not follow Jaynes math fully and there are some papers going back and forth on some of his assumptions, so the mathematical underpinnings may not be as strong as we would like.

I don't know, Intuitively you should be able to ground the agent stuff in information theory, because the rules they put forwards are the same, Jaynes also has a chapter on decision theory where he makes the wonderful point that the utility function is way more arbitrary than a prior, so you might as well be Bayesian if you are into inventing ad hoc functions anyway.

Ahh, I know that is a first year course for most math students, but only math students take that class :), I have never read an analysis book :), I took the applied path and read 3 other bayesian books before this one, so I taught the math in this books were simultaneously very tedious and basic :)

If anyone relies on tags to find posts, and you feel this post is missing a tag, then "Tag suggestions" will be much appreciated

That surprising to me, I think you can read the book two ways, 1) you skim the math, enjoy the philosophy and take his word that the math says what he says it says 2) you try to understand the math, if you take 2) then you need to at least know the chain rule of integration and what a delta dirac function is, which seems like high level math concepts to me, full disclaimer I am a biochemist by training, so I have also read it without the prerequisite formal training. I think you are right that if you ignore chapter 2 and a few sections about partition functions and such then the math level for the other 80% is undergraduate level math

2Iknownothing
I'd also read Elementary Analysis before

crap, you are right, this was one of the last things we changed before publishing because out previous example were to combative :(.

I will fix it later today.

I think this is a pedagogical Version of Andrew Gelmans shrinkage Triology

The most important paper also has a blog post, The very short version is if you z score the published effects, then then you can derive a prior for the 20.000+ effects from the Cochrane database. A Cauchy distribution fits very well. The Cauchy distribution has very fat tails, so you should regress small effects heavily towards the null and regress very large effects very little.

Here is a fun figure of the effects, Medline is published stuff, so no effects between -2 and 2 as they wo... (read more)

SR if you can only read one, if you do not expect to do fancy things then ROS may be better as it is very good and explains the basics better. The logic of Science should be your 5th book and is good goal to set, The logic of Science is probably the rationalist bible, much like the real bible everybody swears by it but nobody has read or understood it :)

Thanks for the reply, 3 seams very automatable, record all text before the image, if that's 4 minuts then then put the image in after 4 min. But i totally get that stuff is more complicated than it initially seems, keep up the good work!

I agree tails are important, but for callibration few of your predictions should land in the tail, so imo you should focus on getting the trunk of the distribution right first, and the later learn to do overdispersed predictions, there is no closed form solution to callibration for a t distribution, but there is for a normal, so for pedagogical reasons I am biting the bullet and asuming the normal is correct :), part 10 in this series 3 years in the future may be some black magic of the posterior of your t predictions using HMC to approximate the 2d posterior of sigma and nu ;), and then you can complain "but what about skewed distributios" :P

-1TLW
...which then systematically underestimates long-tail risk, often significantly.

The text to speech is phenomenal!, Only math and tables suck

Suggestions for future iterations:

  1. allow filtering of the RSS by hacking the url, for example ?sources=LW,EA&quality=curated would give the curated posts from only EA and LW ignoring alignment forum
  2. somehow allow us to convert our "read later" to an RSS :)
  3. when there are figures in the post, then put them in the podcast "image" like SolenoidEntity does for the Astral Codex 10 podcast
  4. put the text in the show notes so we can pause and look at tables
2KatWoods
Thanks for the suggestions! 1. We're working on setting up forum-specific channels. There's just a technical issue we're working on, then it'll happen. 2. Check out Tayl! It does a little what you're looking for.  And you can also use @VoiceReader on Android and NaturalReader on Chrome.  3. Interesting idea! We want this to stay automated so we can focus on other projects and this seems likely to be hard to automate. Will look into it though! 4. This is probably the most requested feature! We're working on it, but it's proving more difficult than we'd expected. Stay tuned. 

It would be nice if you wrote a short paragraph for each link, "requires download", "questions are from 2011", or you sorted the list somehow :)

Yes, You can change future by being smarter and future by being better calibrated, my rule assumes you don't get smarter and therefore have to adjust only future .

If you actually get better at prediction you could argue you would need to update less than the RMSE estimate suggests :)

I agree with both points

If you are new to continuous predictions then you should focus on the 50% Interval as it gives you most information about your calibration, If you are skilled and use for example a t-distribution then you have for the trunk and for the tail, even then few predictions should land in the tails, so most data should provide more information about how to adjust , than how to adjust

Hot take: I think the focus 95% is an artifact of us focusing on p<0.05 in frequentest statistics.

Our ability to talk past each other is impressive :)

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].

Yes this is almost the assumption I am making, the general point of this post is to assume that all your predictions follow a Normal distribution, with as "guessed" and with a that is different from what you guessed, and then use to get a point estimate for the counterfactual you sho... (read more)

Thanks!, I am planing on writing a few more in this vein, currently I have some rough drafts of:

  • 30% Done, How to callibrate normal predictions
  • defence of my calibration scheme, and an explanation of how metaculus does.
  • 10% Done, How to make overdispersed predictions
  • like this one for the logistic and t distribution.
  • 70% Done, How to calibrate binary predictions
  • like this one + but gives a posterior over the callibration by doing an logistic regression with your predictions as "x" and outcome as "y"

I can't promise they will be as good as this ... (read more)

Yes you are right, but under the assumption the errors are normal distributed, then I am right:

If:

Then Which is much less than 1.

proof:

import scipy as sp

x1 = sp.stats.norm(0, 0.5).rvs(22 * 10000)
x2 = sp.stats.norm(0, 1.1).rvs(78 * 10000)
x12 = pd.Series(np.array(x1.tolist() + x2.tolist()))
print((x12 ** 2).median())
3SimonM
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 X∼N(0,1) then yes, I agree the median ofX2 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 you're new claim is "[assuming] the errors are a mixture of normal distributions, centered at 0", which okay, fine that's probably true, I don't care enough to check because it seems a bad assumption to make. More importantly, there's a more fundamental problem with your post. You can't just take some numbers from my post and then put them in a different model and think that's in some sense equivalent. It's quite frankly bizarre. The equivalent model would be something like: p∼Bern(0.78) σ∼p⋅N(1.1,ε)+(1−p)∼N(0.5,ε)

I am making the simple observation that the median error is less than one because the mean squares error is one.

2SimonM
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)

That's also how I conseptiolize it, you have to change your intervals because you are to stupid to make better predictions, if the prediction was always spot on then sigma should be 0 and then my scheme does not make sense

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

1SimonM
I assume you're making some unspoken assumptions here, because 02+22>12 is not enough to say that. A naive application of Chebyshev's inequality would just say that E(X2)=1,E(X)=0⇒P(X≤1)≤1. 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 squared error 1) then you'd find "most" errors are more than 1.

I agree, most things is not normal distributed and my callibrations rule answers how to rescale to a normal. Metaculus uses the cdf of the predicted distribution which is better If you have lots of predictions, my scheme gives an actionable number faster, by making assumptions that are wrong, but if you like me have intervals that seems off by almost a a factor of 2, then your problem is not the tails but the entire region :), so the trade of seems worth it.

2SimonM
You keep claiming this, but I don't understand why you think this

Agreed, More importantly the two distribution have different kurtosis, so their tails are very different a few sigmas away

I do think the Laplace distribution is a better beginner distribution because of its fat tails, but advocating for people to use a distribution they have never heard of seems like a to tough sell :)

My original opening statement got trashed for being to self congratulatory, so the current one is a hot fix :), So I agree with you!

Me to, I learned about this from another disease and taught, that's probably how it works for colorblindness as well.

I would love you as a reviewer of my second post as there I will try to justify why I think this approach is better, you can even super dislike it before I publish if you still feel like that when I present my strongest arguments, or maybe convince me that I am wrong so I dont publish part 2 and make a partial retraction for this post :). There is a decent chance you are right as you are the stronger predictor of the two of us :)

6SimonM
I'd be happy to.

Can I use this image for my "part 2" posts, to explain how "pros" calibrate their continuous predictions?, And how it stacks up against my approach?, I will add you as a reviewer before publishing so you can make corrections in case I accidentally straw man or misunderstand you :)

I will probably also make a part 3 titled "Try t predictions" :), that should address some of your other critiques about the normal being bad :)

Note 1 for JenniferRM: I have updated the text so it should alleviate your confusion, if you have time, try to re-read the post before reading the rest of my comment, hopefully the few changes should be enough to answer why we want RMSE=1 and not 0.
Note 2 for JenniferRM and others who share her confusion: if the updated post is not sufficient but the below text is, how do I make my point clear without making the post much longer?

With binary predictions you can cheat and predict 50/50 as you point out... You can't cheat with continuous predictions as ther... (read more)

3JenniferRM
When I google for [Bernoulli likelihood] I end up at the distribution and I don't see anything there about how to use it as a measure of calibration and/or decisiveness and/or anything else. One hypothesis I have is that you have some core idea like "the deep true nature of every mental motion comes out as a distribution over a continuous variable... and the only valid comparison is ultimately a comparison between two distributions"... and then if this is what you believe then by pointing to a different distribution you would have pointed me towards "a different scoring method" (even though I can't see a scoring method here)...  Another consequence of you thinking that distributions are the "atoms of statistics" (in some sense) would (if true) imply that you think that a Brier Score has some distribution assumption already lurking inside it as its "true form" and furthermore that this distribution is less sensible to use than the Bernoulli? ... As to the original issue, I think a lack of an ability, with continuous variables, to "max the calibration and totally fail at knowing things and still get an ok <some kind of score> (or not be able to do such a thing)" might not prove very much about <that score>? Here I explore for a bit... can I come up with a N(m,s) guessing system that knows nothing but seems calibrated? One thought I had: perhaps whoever is picking the continuous numbers has biases, and then you could make predictions of sigma basically at random at first, and then as confirming data comes in for that source, that tells you about the kinds of questions you're getting, so in future rounds you might tweak your guesses with no particular awareness of the semantics of any of the questions... such as by using the same kind of reasoning that lead you to concluding "widen my future intervals by 73%" in the example in the OP. With a bit of extra glue logic that says something vaguely like "use all past means to predict a new mean of all numbers so far" t

The big ask is making normal predictions, calibrating them can be done automatically here is a quick example using google sheets: here is an example

I totally agree with both your points, This comment From a Metaculus user have some good objections to "us" :)

I am sorry if I have straw manned you, and I think your above post is generally correct. I think we are cumming from two different worlds.

You are coming from Metaculus where people make a lot of predictions. Where having 50+ predictions is the norm and the thus looking at a U(0, 1) gives a lot of intuitive evidence of calibration.

I come from a world where people want to improve in all kids of ways, and one of them is prediction, few people write more than 20 predictions down a year, and when they do they more or less ALWAYS make dichotomous predictions. I ... (read more)

6SimonM
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 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. I disagree that's all you propose. As I said in an earlier comment, I'm broadly in favour of people making continuous forecasts as they convey more information. You paired your article with what I believe is broadly bad advise around analysing those forecasts. (Especially if we're talking about a sample of ~20 forecasts)

TLDR for our disagreement:

SimonM: Transforming to Uniform distribution works for any continuous variable and is what Metaculus uses for calibration
Me: the variance trick to calculate from this post is better if your variables are form a Normal distribution, or something close to a normal.
SimonM: Even for a Normal the Uniform is better.

6SimonM
I disagree with that characterisation of our disagreement, I think it's far more fundamental than that. 1. I think you misrepresent the nature of forecasting (in it's generality) versus modelling in some specifics 2. I think your methodology is needlessly complicated 3. I propose what I think is a better methodology To expand on 1. I think (although I'm not certain, because I find your writing somewhat convoluted and unclear) that you're making an implicit assumption that the error distribution is consistent from forecast to forecast. Namely your errors when forecasting COVID deaths and Biden's vote share come from some similar process. This doesn't really mirror my experience in forecasting. I think this model makes much more sense when looking at a single model which produces lots of forecasts. For example, if I had a model for COVID deaths each week, and after 5-10 weeks I noticed that my model was under or over confident then this sort of approach might make sense to tweak my model.  To expand on 2. I've read your article a few times and I still don't fully understand what you're getting at. As far as I can tell, you're proposing a model for how to adjust your forecasts based on looking at their historic performance. Having a specific model for doing this seems to miss the point of what forecasting in the real world is like. I've never created a forecast, and gone "hmm... usually when I forecast things with 20% they happen 15% of the time, so I'm adjusting my forecast down" (which is I think what you're advocating) it's more likely a notion of, "I am often over/under confident, when I create this model is there some source of variance I am missing / over-estimating?". Setting some concrete rules for this doesn't make much sense to me. Yes, I do think it's much simpler for people to look at a list of percentiles of things happening, to plot them, and then think "am I generally over-confident / under-confident"? I think it's generally much easier for people

I don't know what s.f is, but the interval around 1.73 is obviously huge, with 5-1-0 data points it's quite narrow if your predictions are drawn from N(1, 1.73), that is what my next post will be about. There might also be a smart way to do this using the Uniform, but I would be surprised if it's dispersion is smaller than a chi^2 distribution :) (changing the mean is cheating, we are talking about calibration, so you can only change your dispersion)

Hard disagree, From two data points I calculate that my future intervals should be 1.73 times wider, converting these two data points to U(0,1) I get

[0.99, 0.25]

How should I update my future predictions now?

3SimonM
If you think 2 data points are sufficient to update your methodology to 3 s.f. of precision I don't know what to tell you. I think if I have 2 data point and one of them is 0.99 then it's pretty clear I should make my intervals wider, but how much wider is still very uncertain with very little data. (It's also not clear if I should be making my intervals wider or changing my mean too)

you are missing the step where I am transforming arbitrary distribution to U(0, 1)

medium confident in this explanation: Because the square of random variables from the same distributions follows a gamma distribution, and it's easier to see violations from a gamma than from a uniform, If the majority of your predictions are from a weird distributions then you are correct, but if they are mostly from normal or unimodal ones, then I am right. I agree that my solution is a hack that would make no statistician proud :)

Edit: Intuition pump, a T(0, 1, 100) obviou... (read more)

4SimonM
I am absolutely not missing that step. I am suggesting that should be the only step. (I don't agree with your intuitions in your "explanation" but I'll let someone else deconstruct that if they want)

changed to "Making predictions is a good practice, writing them down is even better."

does anyone have a better way of introducing this post?

2Less_Random
Overall great post: by retrospectively evaluating your prior predictions (documented so as to avoid one's tendency to 'nudge' your memories based on actual events which transpired) using a 'two valued' Normal distribution (guess and 'distance' from guess as confidence interval), rather than a 'single-valued' bernoulli/binary distribution (yes/no on guess-actual over/under), one is able to glean more information and therefore more efficiently improve future predictions. That opening statement, while good and useful, does come off a little 'non sequitur'-ish. I urge to find a more impactful opening statement (but don't ahve a recommendation, other than some simplification resulting from what I said above).

(Edit: the above post has 10 up votes, so many people feel like that, so I will change the intro)

You have two critiques:

  1. Scott Alexander evokes tribalism

  2. We predict more than people outside our group holding everything else constant

  3. I was not aware of it, and I will change if more than 40% agree

Remove reference to Scott Alexander from the intro: [poll]{Agree}{Disagree}

  1. I think this is true, but have no hard facts, more importantly you think I am wrong, or if this also evokes tribalism it should likewise be removed...

Also Remove "We rationalists... (read more)

[This comment is no longer endorsed by its author]Reply

This is a good point, but you need less data to check whether your squared errors are close to 1 than whether your inverse CDF look uniform, so if the majority of predictions are normal I think my approach is better.

The main advantage of SimonM/Metaculus is that it works for any continuous distribution.

SimonM100

you need less data to check whether your squared errors are close to 1 than whether your inverse CDF look uniform

I don't understand why you think that's true. To rephrase what you've written:

"You need less data to check whether samples are approximately N(0,1) than if they are approximately U(0,1)"

It seems especially strange when you think that transforming your U(0,1) samples to N(0,1) makes the problem soluble.

Agreed 100% on 1) and with 2) I think my point is "start using the normal predictions as a gate way drug to over dispersed and model based predictions"

I stole the idea from Gelman and simplified it for the general community, I am mostly trying to raise the sanity waterline by spreading the gospel of predicting on the scale of the observed data. All your critiques of normal forecasts are spot on.

Ideally everybody would use mixtures of over-dispersed distributions or models when making predictions to capture all sources of uncertainty

It is my hope that by ed... (read more)

You could make predictions from a t distribution to get fatter tails, but then the "easy math" for calibration becomes more scary... You can then take the "quartile" from the t distribution and ask what sigma in the normal that corresponds to. That is what I outlined/hinted at in the "Advanced Techniques 3"

Good Points, Everything is a conditional probability, so you can simply make conditional normal predictions:

Let A = Biden alive

Let B = Biden vote share

Then the normal probability is conditional on him being alive and does not count otherwise :)

Another solution is to make predictions from a T-distribution to get fatter tails. and then use "Advanced trick 3" to transform it back to a normal when calculating your calibration.

I think this was by parents, so they are forgiven :), your story is pretty crazy, but there is so much to know as a doctor that most becomes rules of thumbs (maps vs buttons) untill called out like you did

fair point. I think my target audience is people like me who heard this saying about colorblindness (or other classical Mendelian diseases that runs in families)

I have added a disclaimer towards the end :)

I am not sure I follow, I am confused about whether the 60/80 family refers to both parents, and what is meant by "off-beat" and "snap-back", I am also confused about what the numbers mean is it 60/80 of the genes or 60/80 of the coding region (so only 40 genes)

3Slider
60 trait supporting genes out of 80 locations that could support it. I am worried that the main finding is misleading because it is an improper application of spherical cow thinking to a concept that oriented to dealing with messiness.

I totally agree, technically it's a correct observation, but it's also what I was taught by adults when I asked as a kid, and therefore I wanted to correct it as the real explanation is very short and concise.

6localdeity
Ah, that explains it.  Adults are often not very good at explaining science to kids.  And I'd guess the adults in question might not have known that colorblindness was X-linked, even if they were paid to teach science; I think I'd only be surprised by that ignorance in K-12 education if a teacher chose to present the subject of colorblind genetics to the class. I once had a doctor (I'd guess in her early thirties) who, in a discussion of male-pattern baldness, mentioned the mother's father as the best data point—which means it must be X-linked, because otherwise the father's father would be an equally good data point (not to mention the father, if old enough).  I said, "So, it's X-linked, then."  She said, "No, it's not X-linked".  I stated the above logic.  She didn't comment on it, but consulted her computer system, and reported that there were five genes found to be associated with male-pattern baldness, some on the X chromosome and some not.

That is hard to believe, you seem so smart at the UoB discord and your podcast :), thanks for sharing

The University of Bayes Discord (UoB) has study groups for Bayesian statistics which might be relevant to you. The newest study group is doing Statistical Rethinking 2022 as the lectures get posted to YouTube. It requires less math than you have demonstrated in your post.

If you want a slightly more rigors path to Bayesian statistics, then I would advice to read Lambert or Gelman See here for more info.

If you want to take the mathematician approach and lean probability theory first, then the book Probability 110 by Blitzstein is pretty good, the study group... (read more)

Totally agree, it's also Christians critique of the idea :)... Maybe it could be relevant for aliens on a smaller planet as they could leave their planet more easily, and would thus be less advanced than us when we become space faring :)... Or a scifi where the different tech trees progress different, like stram punk

Then maybe it only work for harem anime in space :)

A lot of your latex is not rendered correctly...

Agreed, but then you don't get cool space amazons :). It could be an extra fail safe mechanism :)

2ChristianKl
You can get rid of men entirely and then have your space amazons.

Good Point, In principle the X chromosome already has this issue when you get it from your farther, if the A chromosome is simply a normal X chromosome with an insertion of a set of proteins that blocks silencing, then you can still have recombination, if we assume the Amazon proteins are all located in the same LD region then mechanically everything is as in the post, but we do not have the Muller's ratchet problem

Also the A only recombines with X as AY is female and therefore never mates with an AX or AY

When the space ship lands there is a 1% chance that no males are among the first 16 births ()

Luckily males are firtile for longer so if the second generation had no men the first generation still works

If the A had a mutation such that AX did not have 50% chance of passing on a A, then the gender ratio would be even more extreme, if the last man dies the a AY female could probably artificially incriminate a female.

You can update the matrix and do the for product to see how those different rules pan out, if you have a specific ratio you want to try then ... (read more)

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