Data Scientist

# Jan Christian Refsgaard(Jan Christian Refsgaard)

I think the above is accurate.

I disagree with the last part, but it has two sources of confusion

Frequentists vs Bayesian is in principle about priors but in practice about about point estimates vs distributions

Good Frequentists use distributions and bad Bayesian use point estimates such as Bayes Factors, a good review is this is https://link.springer.com/article/10.3758/s13423-016-1221-4

But the leap from theta to probability of heads I think is an intuitive leap that happens to be correct but unjustified.

Philosophically then the posterior predictive is actually frequents, allow me to explain:

Frequents are people who estimates a parameter and then draws fake samples from that point estimate and summarize it in confidence intervals, to justify this they imagine parallel worlds and what not.Bayesian are people who assumes a prior distributions from which the parameter is drawn, they thus have both prior and likelihood uncertainty which gives posterior uncertainty, which is the uncertainty of the parameters in their model, when a Bayesian wants to use his model to make predictions then they integrate their model parameters out and thus have a predictive distribution of new data given data*. Because this is a distribution of the data like the Frequentists sampling function, then we can actually draw from it multiple times to compute summary statistics much like the frequents, and calculate things such as a “Bayesian P-value” which describes how likely the model is to have generated our data, here the goal is for the p-value to be high because that suggests that the model describes the data well.

*In the real world they do not integrate out theta, they draw it 10.000 times and use thous samples as a stand in distribution because the math is to hard for complex models

Regarding reading Jaynes, my understanding is its good for intuition but bad for applied statistics because it does not teach you modern bayesian stuff such as WAIC and HMC, so you should first do one of the applied books. I also think Janes has nothing about causality.

Given 1. your model and 2 the magical no uncertainty in theta, then it’s theta, the posterior predictive allows us to jump from infrence about parameters to infence about new data, it’s a distribution of y (coin flip outcomes) not theta (which describes the frequency)

In Bayesian statistics there are two distributions which I think we are conflating here because they happen to have the same value

The posterior describes our uncertainty of , given data (and prior information), so it’s how sure we are of the frequency of the coin

The posterior predictive is our prediction for new coin flips given old coin flips

For the simple Bernoulli distribution coin example, the following issue arise: the parameter , the posterior predictive and the posterior all have the same value, but they are different things.

Here is an example were they are different:

Here was not a coin but the logistic intercept of some binary outcome with predictor variable x, let’s imagine an evil Nazi scientist poisoning people, then we could make a logistic model of y (alive/dead) such as , Let’s imagine that x is how much poison you ate above/below the average poison level, and that we have , so on average half died

Now we have:

the value if we were omniscient

The posterior of because we are not omniscient there is error

Predictions for two different y with uncertainty:

Does this help?

I will PM you when we start reading Jaynes, we are currently reading Regression and other stories, but in about 20 weeks (done if we do 1 chapter per week) there is a good chance we will do Jaynes

Uncertainty is a statement about my brain not the real world, if you replicate the initial conditions then it will always land either Head or Tails, so even if the coin is “fair” , then maybe . the uncertainty comes form be being stupid and thus being unable to predict the next coin toss.

Also there are two things we are uncertain about, we are uncertain about (the coins frequency) and we are uncertain about , the next coin toss

I may be to bad at philosophy to give a satisfying answer, and it may turn out that I actually do not know and am simply to dumb to realize that I should be confused about this :)

There is a frequency of the coin in the real world, let’s say it has

Because I am not omniscient there is a distribution over it’s parameterized by some prior which we ignore (let’s not fight about that :)) and some data x, thus In my head there exists a probability distribution

The probability distribution on my head is a distribution not a scaler, I don’t know what is but I may be 95% certain that it’s between 0.4 and 0.6

I think there are problems with objective priors, but I am honored to have meet an objective Bayesian in the wild, so I would love to try to understand you, I am Jan Christian Refsgaard on the University of Bayes and Bayesian conspiracy discord servers. My main critique is the ‘in-variance’ of some priors under some transformations, but that is a very weak critique and my epistemology is very underdeveloped, also I just bought Jaynes book :) and will read when I find a study group, so who knows maybe I will be an objective Bayesian a year from now :)

If I have a distribution of 2 kids and a professional boxer, and a random one is going to hit me, then argmax tells me that I will always be hit by a kids, sure if you draw from the distribution only once then argmax will beat the mean in

^{2}⁄_{3}of the cases, but its much worse at answering what will happen if I draw 9 hits (argmax=nothing, mean=3hits from a boxer)This distribution is skewed, like the beta distribution, and is therefore better summarized by the mean than the mode.

In Bayesian statistics argmax on sigma will often lead to sigma=0, if you assume that sigma follows a exponential distribution, thus it will lead you to assume that there is no variance in your sample

The variance is also lower around the mean than the mode if that counts as a theoretical justification :)

I think argmax is not the way to go as the beta distribution and binomial likelihood is only symmetric when the coin is fair, if you want a point estimate the mean of the distribution is better, which will always be closer to

^{50}⁄_{50}than the mode, and thus more conservative, you are essentially ignoring all the uncertainty of theta and thus overestimating the probability.

Disclaimer: Subjective Bayesian

Here is how we evil subjective Bayesian think about it

**Prior:**Lets imagine two people, Janes and an Alien, Janes knows that most coins are fair and has a Beta(20, 20) prior, the alien does not know this, and puts the ‘objective’ Beta(1, 1) prior which is uniform for all frequencies.

**Data:**The data comes up 12 heads and 8 tails

**Posterior:**Janes has a narrow posterior Beta(32, 28) and the alien a broader Beta(13, 9), Janes posterior is also close to

^{50}⁄_{50}if Janes does not have access to the data that formed his prior or cannot explain it well, then what he believes about the coin and what the alien believes about the coin are both ‘rational’, as it is the posterior from their personal priors and the shared data.

**How to think about it:**Janes can publish a paper with the Beta(13, 9) posterior, because that is what skeptical people with weak priors will believe, while himself believing in a Beta(32, 28)

To make it more concrete Pfizer used a Beta(0.7, 1) prior for their COVID vaccine, but had they truly belied that prior they would have gone back to the drawing instead of starting a phase 3 trial, but the FDA is like the alien in the above example, with a very broad prior allowing most outcome, the Pfizers scientists are like Janes, we have all this data suggesting it should work pretty well so they may have believed in Beta(5, 15) or whatever

The other thing to notice is the coins frequency is a distribution and not an scalar because they are both unsure about the ‘real’ frequency

Does this help or am I way off?

The probability is an external/physical thing because your brain is physical, but I take your point.

I think the we/our distinction arises because we have different priors

Cholera is the devil!

The National Center for Biotechnology Information has a Taxonomy database.

Q: What do you think taxid=666 is?

A: Vibrio cholerae, coincidence? I think not!

proof:

https://www.ncbi.nlm.nih.gov/Taxonomy/Browser/wwwtax.cgi?mode=info&id=666

# Jan Christian Refsgaard’s Shortform

I loved that example as well, I have heard it elsewhere described as “The law of small numbers”, where small subsets have higher variance and therefore more frequent extreme outcomes. I think it’s particularly good as the most important part of the Bayesian paragdime is the focus on uncertainty.

The appendix on HMC is also a very good supplement to gain a deeper understanding of the algorithm after having read the description in another book first.

I think we agree and are talking past each other, my original statement was “Most statisticians would agree with you.

**Unless**...”So we agree that there is more power in

^{3}⁄_{5}than^{2}⁄_{3}, and we happen to have divergent intuitions about what random Joe finds most persuasive, my intuition is rather weak so I would gladly update it towards^{3}⁄_{5}sounding more impressive to random people, if you feel strongly about it.Most likely what the marketing folks have done is gotten a list of top 100 runners in different running disciplines and the reported “the most impressive top X in list Y”,

We both agree that the reported statistic is inflated, which is the major thesis, we simply disagree about how much information can be recovered because we have different “impressiveness sounding heuristics”

Good point!

original: Applied Bayesian Statistics—Which book to read?

Applied Bayesian Statistics—Which book should you read?

Literature Review of 5 Applied Bayesian Statistics Books.

Book Review of 5 Applied Bayesian Statistics Books.

I picked 3, if other people have strong feeling feel free to suggest other titles

I disagree, if Reebok produced 64% of all shoes in the world and only

^{3}⁄_{5}of top athletes used them, and this furthermore was the best statistics the marketing department could produce, then it’s strong evidence that they are over hyped.But I think you understood me as saying something different, words are hard :)

If Reebok wanted to report a valid statistic they would report something like 11% of the top 100 wears our shoos, I think a much smaller number than top 100 was picked exactly because that was where the effect was the most exaggerated.

**ChristianKl**share your intuition that^{3}⁄_{5}sounds more impressive than^{2}⁄_{3}. I also agree that reporting top 4 would seem even more fishy, though it could be spun as 75% of quarter finalists in knockout tournament sports.

I have written a review of the 5 most popular Applied Bayesian Statistics books

Where I recommended:

Statistical Rethinking

Up to speed fast, no integrals, very intuitive approach.

Doing Bayesian Data Analysis

This is the easiest book. If your goal is only to create simple models and you aren’t interested in understanding the details, then this is the book for you.

A Student’s Guide to Bayesian Statistics

This book has the opposite focus of the Dog book. Here the author slowly goes through the philosophy of Bayes with an intuitive mathematical approach.

Regression and Other Stories

Good if you want a slower and thorough approach where you also learn the Frequentest perspective.

Bayesian Data Analysis

The most advanced text, very math heavy, best as a second book after reading one or two of the others, unless you are already a statistician.

I will write a post shilling for myself, thanks. I was waiting for the post to be ‘liked’, if it got −10 karma then there would be no use in shilling for it :)

I am one of those people with an half baked epistemology and understanding of probability theory, and I am looking forward to reading Janes. And I agree there are a lot of ad hocisms in probability theory which means everything is wrong in the logic sense as some of the assumptions are broken, but a solid moden bayesian approach has much less adhocisms and also teaches you to build advanced models in less than 400 pages.

HMC is a sampling approach to solving the posterior which in practice is superior to analytical methods, because it actually accounts for correlations in predictors and other things which are usually assumed away.

WAIC is information theory on distributions which allows you to say that model A is better than model B because the extra parameters in B are fitting noice, basically minimum description length on steroids for out of sample uncertainty.

Also I studied biology which is the worst, I can perform experiments and thus do not have to think about causality and I do not expect my model to acout for half of the signal even if it’s ‘correct’