The biggest beef in statistics explained

Non-clickbait title: Explaining the differences between frequentist and Bayesian statistics (with analyses)

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[berjonah]

I tend to lean toward the Bayesian approach for two reasons: It tends to be easier to build up complicated models using conditional latent variables, and the prior distribution gives a way to incorporate expert knowledge about a subject. I've worked with many subject matter experts who don't have a firm grasp of statistics, but are very knowledgeable about their own corner of the world. Having the ability to take what amounts to "vibe based reasoning" from them and quantify it using an informative prior distribution gives a lot more power than just using a flat prior.

[dhonantarogundul]

I always interpret frequentist statistics as "static statistics" and Bayesians statistics as "dynamic statistics, " which works well in my field of study, robotics!

[micheldavidovich]

Question about the null hypothesis you selected as I’ve had this is ue come up as well. Why do you select H0) pi = 85%? If you want to make a decision on whether the coffee shop is good or bad, shouldn’t it make more sense to asume pi <= 85%. That would tell you that if you repeated the experiment multiple times, 95% of your confidence levels would hold the true value, and that it would be towards the good side. I do agree the bayes formulation a much more beautiful in any case 😊

[Ethan]

One option for avoiding Bayesian prior-rigging is to simply publish the calculation itself, lacking any prior. Then, the probability is simply some function of the prior, which you could graph or something to visualize it better

[PrParadoxy]

Bayesian view doesn't apply very well to statistical physics. There is a concept called micro canonical ensemble, where we assume the frequency of each event to be equal. From this, we can calculate entropy, and from that one calculate other physical quantities of interest like temperature. In Frequentists' view, no problem arises as the physical system's properties is independent from observers knowledge. Everyone agree on the temperature of the box of a gas. However, in Baysian point of view, if someone (say by prior measurements) have extra knowledge about the system, they would not assign equal probabilities to each individual events, causing them to claim a different entropy, temperature, etc, which would not agree with our actual observation. I have seen some effort to fix this, however, Baysian view is not as natural as this video makes it to be.

[davidarredondo]

Excellent video!! I’m almost done with Bernoulli’s Fallacy myself.

I do want to add that, for what I’ll call “reasonable” priors, the choice of prior doesn’t matter in the long run, as the data will dominate the posterior through the likelihood.

Basically, with Bayesian statistics, we’ll find the truth if we just keep on collecting more data.

Again, thanks for this great summary! I teach both a high school stats course and a high school Bayesian data science course, and this is the best short explanation of the difference I’ve seen. Congrats!

[hughobyrne]

It seems like having factions that say "Pi is determined by geometry: the ratio of a circle's circumference to its diameter" and "Pi is determined by calculus: the limit of an infinite sum (pick your favourite)".

[scepticalchymist]

Idealists will not stop to debate which approach is more valid, pragmatists will just use one or the other depending which one fits best in any given situation.

[Zxymr]

I told my Asian parents that I was Bayesian.
They disowned me.

[AdamKuczynski]

Surely more people have visited the cafe than have left a review? So repeating the experiment and collecting another ~1k reviews from those who simply hadn't written theirs down isn't all that far fetched. Impractical, yes, but entirely possible. The idea of 'repetition' breaks down much more when we think of data we can't really sensibly resample/remeasure (like a country's annual GDP or the employment rate).

[LNVACVAC]

I am not a mathematician but I have knowledge on biostatistics.

The frequentist approach falls short in regards of rare diseases because:

1 - The definition of rare disease is both arbitrary and normative. (less than 1 in 2 thousand).

2 - Most medics, even doctors, are not sufficiently informed about rare diseases.

3 - Differential diagnosis typically is symptom informed and not revolving on the investigation of prevalence of specific causes. (Example: when you go do the medic with a sore throat the medic doesn't ask for a swab before looking for bacterial plaque. However only a subset of individuals with bacterial infection and sore throat will have bacterial plaque when seeing a doctor.)

4 - All these not only create sub notification, typically above 40%, but also the samples of control and unhealthy individuals will not approach infinity.

5 - Complex-Adaptative Natural Entities often behave like Loaded Dice. Not only in prevalence but in appearance/aspect, aggravating the Item 3 problem.

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Mathematicians need to understand these are not transcendental matters and that tools are instrumental, not necessary. This battle is not a first principles contention.

[xenoduck]

Bayesan probability still has the same definition as frequentist probability!!! What you are showing is not a "definition" of probability, it is just Bayes' rule, which says NOTHING of P(A), only of P(B|A). The law of large numbers gives the definition of probability, regardless of what field of maths you study. I feel like this was really misrepresented in the video.

[PerishingTar]

You got my butt with that ad transition 😅

[tahsinahmed]

How would you explain the choice of the likelihood distribution chosen?

[huhuboss]

Why would you risk having a wrong prior when you can simply use the frequentist apprach? Genuine question

[danielkeliger]

Yes, I totally agree that for large sample sizes the two methods basically give the same asnwer. They are also compatible in the sense that Bayesians have their own interpetarion for maximum likelihood and Bayesion methods can be analysed via frequentist language. (Infact it is more natural to understand the limit theorem mention in the video in frequentist terms, in my opinion.)

Still, I want to make some remarks.

Firstly, I’m sort of a prularist. I don’t think probability stands for a single concept. Statements like “what is the probability that this previously unknowm sonnet was written by Shakespeare” can be interpeted in a Bayesian way much more generally, while physical problems (see belowe) makes more sense in the frequentist interpetation. Ultimately, there are many things that satisfied by the Kolmogorov axioms that has nothing to do with randomness. (Say the ratio of votes in an election.) It is possible to do probability theory without reffering to randomness at all.

There are cases when we do actually talk about frequencies in the world. Ergodicity is s good example. Saying things like “if I know the exact inital conditions, I can calculate the exact ratio of times the coins will land on head” and therefore “probabilites are purely epistemic” kind of misses the point. I’m not interested in this very particular initial condition. I want to show that this behaviour when roughly 1/2 of the coin tosses lands on head is typical for most of the initial conditions. This 1/2 number is a property of the system, and it doesn’t describe the mental state of an idealised, rational observer. (With the obvious objection that of cource observations themselves are model dependent, etc.)

Lastly, all the populat interpetations have their own philosophycal problems. I don’t know any interpetation that are not ultimately flawed under greater scrutiny. This is actually very typical when it comes to phylosophical problems. (Think about all the different schools of ethics.) I think I like the propensity interpetation of probability the most, but that is not perfect either.

[alex_zetsu]

In my opinion, these two philosophies can be reconciled by thinking of frequentist statistics as just approaching the problem with a specific prior that is asking "am I X percent confident in posterior outcome A?"

[TwentyNineJP]

Finally I have the vocabulary to describe my philosophical objections to the way that the topic of statistics is often discussed. Probabilities have no place in a world of perfect knowledge; to a hypothetical god, all probabilities would be either 1 or 0, and nothing in between.

It is only our ignorance of outcomes that gives meaning to statistics.

FWIW the Bayesian approach is what I studied in signal analysis. I just didn't realize that the whole of statistics was bifurcated like this.

[tobiaseriksson]

How come we assume everything is a Gaussian or treat things as if they where? Alot of statisticial tests rely on it, but it seems like all of the factors for the test to be valid is always not respected.

[Shantanu_Dixit]

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