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.

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.

When I use machine learning algorithm to predict stuff, is it the bayesian way or the frequentist way ? or something between both or does it really depends on the data distribution or depend of the specific machine learning algorithm ?

Read "A history of mathematical statistics (from 1750 to 1930)" by Anders Hald

You got my butt with that ad transition 😅

Not a statistician, but I do have a take on this... The Bayesian method relies on priors which hamstrings the whole practical purpose of analysis. Instead of debating results, people instead debate priors. It just shifts the whole thing from one frying pan to the other. The simplistic frequentist approach you described is utterly naive. You completely missed the whole concept of random walk. In practice, the most reliable approach to probability is the non-naive version with a large dataset, or a large set of datasets. Random walk is critical to understand for the frequentist approach to make any sense. For example, imagine flipping a balanced coin 4 times (small example, easier to explain) The naive approach would assume that larger datasets tend towards 50% heads, but this doesn't make sense. The probabilities are: 0% heads -> 1/16 25% heads -> 4/16 50% heads -> 6/16 75% heads -> 4/16 100% heads -> 1/16 It's a bell curve centered at 50%. With large data sets, your chance of getting the expected 50% result is only around 6/16, but your chances of getting either 25% or 75% is 8/16... Which means the naive approach is more likely to give an inaccurate result! Random Walk (results steering away) is a huge topic in itself and definitely needs to be accounted for to rely on the frequentist method.

Damn dude, what is the frequency of you hitting the gym? Your arms are BIG

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)".

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!

Today I learned: some people pronounce Bayesian like “beige” rather than “Bayes”

I'm a big predictions markets and forecasting guy so I essentially take Bayes for granted. Typically you look at past frequencies then apply a prior to it, which can get kinda subjective. Bayesian as a word seems so fancy (aka objective) I thought you could never be truly Bayesian. But really its just fancy terminology ppl say to look cool and not be a frequentist dummy who doesn't understand that randomness exists. I never really saw it as a conflict.

Can you talk more about bootstrapping?

I really enjoy your videos and style of teaching. I found this video especially well done and I learned a lot about a subject that's hard to understand (I'm an architect, not a mathematician, but I love this subject). I have been trying to learn statistical concepts for years and from this video I am starting to understand that if you are a mathematician focused on the computation and not knowledgeable about the subject you are studying, a frequentist approach may be more appealing and appropriate. If you are a subject matter expert trying to predict possible future outcomes, a Bayesian approach would be a better fit. You could say it caters to your prejudices, that's fair, but it also allows you to employee your expertise. So perhaps a Bayesian approach is more corruptible, but done properly it seems to have higher potential.

So now we are backing up probability in the single biggest flaw in the history of science, the human bias?

good video, but I'm lost near 10:00

Nonsense. The biggest beef in statistics is the thirders vs the halvers. #thirdersftw!!!!

#stopbayesianhate

Do you use Manim to make your visualizations ? I love how you work through the concepts and keep the canvas as clean as possible. Keep up the great work 🎉

There are frequintists? Really?, i thought they were a story told to scare children.

Hi, did your Masters had wetlab?

Frequentism? Oh you mean Nazi race science math. (Bayes 4 lyfe boiiiiiii)

Please, I'm begging you, distinguish between [1] Bayes' Theorem, which is the thing you talk about at ~4:45 and which frequentists fully endorse (since it follows from the axioms, the ordinary definition of conditional probability, and classical logic), and [2] Bayes' Rule, properly so-called, which is a claim about how degrees of belief or confidences should be updated in light of evidence -- namely, that belief update should be by way of conditioning on one's evidence, i.e. Pr_new(h) = Pr_old(h|e), where e is the new evidence.

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.

Bayesian propaganda 😂

I think you haven't really got the essence of Bayesian theory. You write such things as P(A). That's against the basis of the Bayesian approach. The basis is that you must state specifically the data and grounds on which you rely for the the value of your probability. That means that you should write only such things as P(A|θ), where A denotes the event as usual, and θ denotes the data and grounds on which you rely for the value of your probability. For example, A could denote 'heads' in a single toss, and θ could denote the proposition that the coin is a fair coin so that only heads in a single toss 'A' and taiis in a single toss 'B' are possible and P(A|θ) = P(B|θ). It's called transparency, and just requires you to state you position explicitly. So-called 'Bayes Theorem' isn't the essence of the approach; it's just a theorem that is true for frequentists as well. The essence is the requirement for explicit transparency. The Bayesian approach has nothing to do with subjectivity. The allegation of subjectivity is just frequentist rhetoric intended to badmouth the Bayesian approach. Perhaps I have been unfair in the foregoing by under-recognising what you said, but I ask for a harder line on the inadmissibility of such expressions as P(A).

Great Video, thank you! Regarding 17:05: Would you have some references for the "some" that "have argued that the bootstrap form something similar" would be quite applicable for a current project of mine

It's very arguable if the idea of probability itself is a fundamentally real thing. Like as a mini example the digits of pi behave random by every single metric we know, yet they are determenistic and nothing random is happening. The ultimate goal of probability is modeling unknown outcomes and that can be done in many ways. So there is no true right option, all we care for is how accurate we can predict things and how interpretable it is to us. (ps in my eyes Bayesian feels more true to real life and my thinking)

New found love for statistics....Thank you so much!!

I guess ,probability is derived from geometric property of our microspace (like general relative theory is derived from timespace geometry). So frequentist approach is more relevant.

You just gained a subscriber love your content 🌿🌿🌿

One of the advantages of the Bayesian approach is it feels more "natural" to incorporate non-quantitative evidence into your calculations. For example it's pretty easy in a frequentists analysis to calculate the odds of rolling a 6 on a d6 if you have rolled it a hundred times and can see the distribution of prior outcomes (fair or loaded). If instead you tell them we have no prior data but there's double sided tape on the 1 side you can easily "swag" that prior with a Bayesian approach and get better results. I've never actually seen a calculable advantage to either view but if you start fudging some numbers using a frequentist approach it just feel like you are doing something wrong... I don't actually think there is a difference or if there is I clearly don't understand it.

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

As usual for a bayesian video, there is much bias towards complexification. First, the test should be one sided: you requested at least 85%, so please have the courtesy to do the correct one. That divides p value by 2 from scratch. Then, you do not need a confidence interval at all, you have p value. What the test tells you is that from a sample of 1074 people, there was a probability of 0.27% to get the data you got if anyone was puting 4 or 5 stars _less_ than 85% of the time (by the way, this is how you got the 99.7% "that only Bayesian gives you", supposedly....). This is the frequentist approach, and it deals with facts and makes two assumptions: independance of choices from users and validity of CLT. Then from that p value, you can do what you want, you are not even obliged to do anything, because so far you only collected data and did maths. Once the computation is done, you can _finally_ go philosophical and decide you do not live in a universe where you got unlucky to be in the 0.27%. There is no binomial, no beta, no prior, no "I don't have an idea of my prior, so I'll use uniform distribution but I will call it Beta(1,1)", no some god of philosophy told me that "no idea" meant the existence of a uniform distribution in the realm of ideas, etc... Frequentist works with facts and try, at least when they're not psychologists or marketers, to be rigorous, not forgetting the assumptions they made. They uses stats to falsify theories, and they don't put probabilities on theories which rermain true or false. Bayesians do decision making, using a tool that always works, always getting an answer whatever was the question they had. It is very good for investors who want to use some maths and have a magical tool that allow them to propose a strategy with some appearance of seriousness, and it will work whenever they were lucky with their priors. But at the end of the day, either your posterior "probability" depends a lot on your priors, and you only put a number on your feelings, or it doesn't and you didn't need to go Bayesian. Frequentists don't deal with philosophy. Bayesian do and must.

Frankly a ridiculous theorem. Human interpretation should have no merit in the actual probability of an event; otherwise simple probabilistic tricks such as the Monty Hall Problem would never be solved. "Frequentist" interpretation doesn't mean you need to throw all context related to the event to the side and calculate it in a vacuum. That is simply bad definitions of statistical problems.

Im not a statiatician, but it feels like you are strawmanning their argument. Instead of a second 1000 review experiment, you could have made 2 different selections of half and you do have 2 different experiments. Thw repeated experiment is the 1000 different reviews. It makes you lose credibility when you do this.

Oh boy, we're gonna talk about Laplace's rule of succession! (<- wrong) Anyway. I really don't like the underlying assumption that all the data is equally relevant - maybe, for example, the cafe changed beans six months ago, and 50 of your 1000 reviews are very important, and 950 are pulling the review in an unrelated direction - is there an approach that would be able to infer things like this mathematically, or is it data weighting always part of the input? When I think about trying to pull valuable metadata from, say, a survey, it seems like it would be critically important not to let filtering and weighting be human decisions.

Btw - my arguments with statistics do not mean they are useless. Rather, they are frequently and all too easily abused (intentionally or unintentionally). In every instance, the use of statistics as applied to real world analyses must be critically analyzed and scrutinized, starting with the assumptions, presumptions and desires and relative ignorance of those involved. Even when all of that is fair, statistics often goes wildly away from truth or reality. And researchers often fail to apply even the most basic critical analyses to the results. Is the population a single population? Is the population linearly, triangularly, normally, poison or other distributed? Are there hidden variables? Is the data the result of stochastic events acting on stochastic events? Do the results violate sanity? Do the results suggest results outside the bounds of the analysis? Is the thesis or hypothesis that resulted in the data gathered biased in its own rights? Etc...

Both frequently lead to bogus answers by never understanding the problem in the first place (in dozens of different ways), and by wrongly conflating chance, risk, and probability of many different flavors, and by presuming assumptions that have no basis in reality. Lies, damnable lies, and statistics - the most damnable lies of all. Beyond all of this, statistical analysis is subject to thousands of human flaws in cognition, in emotion, in biases, and more. Statistical analysis often then gets subsumed into bogus hierarchical fault tree analysis that pile error upon error, and/or supporting bogus and erroneous multi-function weighted attribute analysis, and/or conflating cost-benefit analysis as dispassionate, fair and/or meaningful, and/or truncating high-consequence low-probability events, and/or utterly missing consequence analysis and vulnerability assessment, and/or utterly failing to understand and properly apply "Safety Culture", and wrongly substituting a wildly erroneous belief that a misunderstanding about the meanings of the two words is a substitute fir understanding and properly application of the phrase. Oh yes, AND confusing or reversing cause and effect, or assuming causal linkage when no linkage exists. Said more simply: a pox on both.

So when you say the probability of this happening, you are a frequentist and if you say it you are bayesian.

The "subjective" thing is entirely your invention. I've never heard of inventing the prior probabilities when using Bayes' rule. You compute the probabilities as best as you can and then use those in Bayes' formula.

You did a two-sided test when a one-sided test was needed. The actual pvalue was a half of what you got. Your H0 should have been pi<0.85 and H1 should've been pi>=.85

My statistics teacher refused to teach Bayesian inference because you got a supposedly improved probably estimate with no additional information. Information theory is important in my work. Generally, a frequencist approach works for me. Adding information that isn't there by using an a priori model can be a problem.

"C.I does n't tell us if it contains the true value of PI or not, you can only know that if you repeated the experiment multiple times then most of them will. " Can you explain this statement I didn't get it.

the only thing i didn't understand is the 0.05 i mean why we chose it ? what is special about 0.05 exactly? is it used for all problems or it can be calculated to fit the situation better ? using one constant assumption for all situations does't make sense i hope anyone answer my question

The idea of a repeated experiment showcases the inherent variability in the parameter estimate, i.e. the sampling distribution. A frequentist assumes that there could be a different data set borne by the same invisible data-generating process (law). Bayesians tend to jump onto data matrices as if those n=200 observations were the one and only realisation possible, without other hypothetical scenarios occurring, as if there were no Heisenberg principle or quantum uncertainty. The frequentist approach reflects the randomness of Nature and unobservability of hypothetical outcomes better: ‘it could have been otherwise’. Finally, Bayesians often make ridiculous distributional claims: ‘assuming the prior normal distribution, the posterior distribution of the linear regression slope estimator is precisely Student with n=198 degrees of freedom', whilst frequentists are much more careful about heteroskedasticity, calibration, coverage probability, and Bartlett correction, which are essential to control the false discovery rate: ‘there is some unknown law, but we can compute some functionals thereof regardless of the joint and marginal distributions, as long as enough finite moments exist for the WLLN and CLT to work’.

O love this guy

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

To understand the beta distribution: 1. Imagine you are sending rockets to an alien planet to see what portion of the surface is covered in water. 2. You can send probes that hit the ground, are destroyed instantly, but send back whether they landed on water or land. 3. Let's say you send down a probe and it says it hit land. 4. If the entire planet were water, there is a 0% chance the probe would say land. If the planet were 100% land, there is a 100% chance of it happening. If you plot the percent of land on the planet on the x axis, and the relative probability that the probe says land, you get a line from 0,0 to 1,1. You can image the opposite being true if it said water: a line from 0,1 to 1,0 5. If you send another probe and it says water, then you can combine two plots. multiply the land plot by the water plot because at each possible percent of land on the planet, the probabilities are being combined. You'll end up with a parabola. 6. Keep multiplying the right plot if the probe says water or land, slowly you'll get a bell curve where the peak is at the ratio of land and water probes. This is what the beta distribution is.

20:04 - "You can have strange priors, but you're going to have to justify them with evidence." But in that case, they're not really a subjective prior at all, are they? If they're properly evidence-based, then they're objective, surely? And in that case, the fundamental objective/subjective difference that you'd previously described is no longer there. An objective prior followed by an objective analysis gives an objective result, and a subjective prior followed by an objective analysis gives a result that is to some extent not evidence-based!

If anyone is interested in if there is an objective way to pick a prior probability distribution, you do it with something called "maximum entropy". And the entropy they refer to is the same one the physicists talk about.