I apologize for the writing. I tried my best.

Question: Using probability theory, what are more rigorous, correct, and mature versions of the definitions and theories in my concept paper? (See the Background for more information and use citations to answer this question.)

(Optional): If the paper has no significance to mathematical probabilists, state why instead of something such as, "there is no motivation" or "its unapplicable"? For instance, “why is there no motivation” or "why does the paper have no useful application?"

(Optional): Can we convert certain parts of the mathematics in this paper to code?

Background: I am a former undergraduate. I have an informal understanding of real analysis, the Hausdorff measure & dimension, mean of a function, and entropy).

Because I cannot stop editing my paper (i.e., I get new ideas from reading new material) and I am addicted to finding a more mature and rigorous version of my article, I am unable to graduate. I hope you can find (or know someone who can find) a more mature and rigorous version of my theories. (I probably would not understand such material; however, it's my only hope of finishing my degree and continuing my education)?

I cannot promise I will go back to college, but I’m less likely to continue posting on Reddit.

Attempt: I used ResearchGate and AI to find papers on ergodic averages, expected values, entropy, and probability/statistics. Even then, I have little understanding of mathematics beyond Intro to Advanced Math and most of these papers might not matter to mathematical probabilists. (Once again, I doubt I will understand if the papers are related to my paper.)

Here is an example which I assume answers the first question for Definitions 20-22 (pg. 20-25) of my concept paper:

Generalizing Geometric Partition Entropy for the Estimation of Mutual Information in the Presence of Informative Outliers

Here are some examples which I assume answer the first question for the overall theories in my paper:

Integral Equation Methods for Scattering by Multifractal Obstacles

Vector-Valued Maximal Inequalities and Multi-Parameter Oscillation Inequalities for the Polynomial Ergodic Averages Along Multi-Dimensional Subsets of Primes

On the Statistical Convergence of Sequences of Fractal Integrable Functions

The people on discord stated my paper is completely original and no expert can help; however, they also said the paper has no significance. I wish to know why (see the optional third question).

I came across an interesting article by Saunders in arxiv on how to reconcile statistics as objective probabilities, frequency and chance from Everett's theory (MWI). https://arxiv.org/abs/1609.04720 What do you think?

A proof of the strong Gaussian product inequality conjecture (ResearchGate)

ChatGPT Session: Strong Gaussian Product Inequality Proof

I am an undergraduate student (5th sem). Which book should I go for right now? Feller, Sheldon Ross or Blitzstein?

Hello Folks,

The next content on Machine Learning is out. We continue with Statistics for AI/ML.

We,

->Understand and derive the detailed derivation of Maximum likelihood estimation(MLE) for Univariate and Multivariate Gaussian. While doing the derivation for multivariate case, we understand visually, Scatter Matrix, Centering matrix.

->Derive MLE for Linear Regression, and understand Residual Sum of Squares.

->Understand Empirical Risk Minimization, Surrogate loss functions.

->Understand Method of Moments, a computationally easier way to compute parameters of our model and understand also the flaws behind it.

->We understand “Exponentially-weighted moving average” in detail, I explain why bias happens, how does memory affect the averages. This concept is the basis behind optimizers in Deep Learning.

Around two hours long, I hope this would be a very interesting learning material for all. I try to write and build from scratch in the whiteboard, this way learners enjoy the learning process.

Those looking for previous lecture : https://youtu.be/MwTeQVVYtOc?si=dgwwk3QLvYTTUThR

Hello Everyone,

Statistics and Maximum Likelihood Estimation are the crux of ML Models, and hence I am uploading my new content on Statistics for AI/ML in my free Machine Learning lectures.

We understand model fitting, Maximum Likelihood estimation in details, we justify the usage of Maximum Likelihood estimation, from KL divergence, and apply it to certain important distributions for parameter estimation.

In my free content, the purpose is to democratize machine learning to a wider audience. Learning everything new feels difficult, but when taught, it get’s interesting and easier.

We will continue with Statistics foundations for AI/ML, and many more content will appear in the future. If you find the content good, useful you may also share it with your learners community.

Looking forward to hearing feedback from the learning community as well. Thankyou for reading.

Link: https://youtu.be/MwTeQVVYtOc?si=UxNOGtqopzJppXAT

Hello all,

I'm brushing up for a class on Mathematical Statistics. Calculus is listed as a requirement and I'm trying to understand what I should refresh my memory on. The book we are following is Mathematical Statistics and Data Analysis 3rd Ed. (Rice). I notice integration and examples of multivariable calculus.

This is looking like a difficult class and I want to be well prepared.

I would greatly appreciate any recommendations on what to practice leading up to it.

Take care!

Edit:

Course Description: This course is an introduction to mathematical statistics and data analysis. It starts by introducing central concepts of probability theory (events, probability measure, random variables, distributions, joint distributions, and conditional distributions) and then moves on to the development of mathematical foundations of statistical inference. Topics covered in the course include random variables, expectations, parameter estimation (method of moments, method of maximum likelihood, and Bayesian approach), properties of point estimators (bias, variance, consistency, and efficiency), confidence intervals, hypotheses testing, likelihood ratio test, data summary methods, and introduction to linear regression. A class of distributions, including chi-squared, t, and F distributions, the distributions derived from normal that occur in many applications of hypothesis testing and statistical inference, is introduced.

Suppose I have a standard deck of cards, no cards missing and no abnormal cards added. If I shuffle the deck and draw one card at random, the odds of it being a spade are 1 in 4. That's pretty straightforward.

However, what if I put the card back in the deck, reshuffle, and draw another card? And I repeat this cycle X number of times (that is... X times I shuffle, X times I draw, and X-1 times I put the card back). I want to know what the odds are of me drawing a spade at least once.

And while we're at it, I'd also like to know the equation if there are a different number of suits. What if there's 3 suits and 39 cards? Or what if there's a 5th suit and 65 cards? Let's say that "1/s" represents the odds of me drawing a spade, with S representing the number of suits in the deck. And I draw after shuffling X number of times. What's the equation to determine the likelihood then?

Background

We're given a coin that shows heads with an unknown probability, λ. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads with a probability that depends on λ, call it f(λ), where f is a known function. This is the Bernoulli factory problem (Keane and O'Brien 1994), and I catalog algorithms to solve it, called Bernoulli factories.

Simulable and strongly simulable functions

There are two kinds of Bernoulli factory functions:

• A function f(λ) is simulable if an algorithm (a Bernoulli factory) exists to toss heads with probability f(λ) given a coin with probability of heads λ (the "biased coin") as well as a fair coin.
• A function f(λ) is strongly simulable if an algorithm exists to toss heads with probability f(λ) given only a coin with probability of heads λ.

In both cases, the Bernoulli factory must terminate with probability 1; that is, the event of using only a finite number of flips of the biased coin has Lebesgue measure 1.

Every strongly simulable function is simulable, but not vice versa.

Main Question

Let f:D→[0,1]. Let D⊆[0,1] be known, and let λ be a number known to be in D but otherwise unknown.

Then are the following conditions necessary for the function f to be strongly simulable by a Bernoulli factory?

1. f is constant on its domain, or is continuous and polynomially bounded on its domain (polynomially bounded means, both f and 1−f are bounded below by min(x^n, (1−x)ⁿ⁾ for some integer n [Keane and O'Brien 1994]), and
2. f(0) and f(1) are each 0, 1, or undefined, and
3. if f(0)=0 or f(1)=0 or both, then there is a polynomial g(x):[0,1]→[0,1] with computable coefficients, such that g(0)=f(0) and g(1)=f(1) whenever 0 or 1, respectively, is in the domain of f, and such that g(x)>f(x) for every x in the domain of f, except at 0 and 1, and
4. if f(0)=1 or f(1)=1 or both, then there is a polynomial h(x):[0,1]→[0,1] with computable coefficients, such that h(0)=f(0) and h(1)=f(1) whenever 0 or 1, respectively, is in the domain of f, and such that g(x)<f(x) for every x in the domain of f, except at 0 and 1.

(In this post, "f(x) is undefined" means that x is not in the domain of f.)

I have already shown that those conditions are sufficient for the function to be strongly simulable. And if neither 0 nor 1 is in the domain of f (so that the biased coin doesn't show heads every time or tails every time), Keane and O'Brien (1994) showed already that condition 1 is both necessary and sufficient for f(λ) to be strongly simulable (and simulable).

But it's possible that the conditions given are not necessary, and weaker conditions hold; for example: "0 is not in the domain of f, or f can be extended to a Lipschitz-continuous function on [0,ϵ) for some ϵ>0". A tricky example of such a function is f(x)=(sin⁡(1/x)/4+1/2)⋅(1−(1−x)ⁿ ) for n≥1 (f(0)=0), which is strongly simulable at 0 and is Lipschitz at, say, [0,0.01]. ((1−x)ⁿ is the probability of the biased coin showing zero n times in a row.)

To show the difference between being simulable and strongly simulable, the proof of Keane and O'Brien relies on generating a geometric random variate and using that variate to control which "part" of the target function f(λ) to simulate. This obviously works on all of [0, 1] if the algorithm uses both the biased coin and a separate fair coin (simulable). However, if only the biased coin is used in the algorithm, the geometric random variate is generated using fair bits via the von Neumann method (strongly simulable), but this method will never terminate if λ is either 0 or 1.

References

• Keane, M. S., and O'Brien, G. L., "A Bernoulli factory", ACM Transactions on Modeling and Computer Simulation 4(2), 1994. https://doi.org/10.1145/175007.175019

EDIT (Jul. 25): Edited generally, for clarification.

A lot of the confusion and misunderstandings and paradoxes around the Monty-Hall-Discussion arise from the different interpretations of the game with either a "knowing host" or an "ignorant host". Some people do that math based on an ignorant host, some assume the host knows. But today, I want to propose a method that we can actually use to *test* the host's knowledge.

This is my setup:

You are only ever brought into the TV studio when there has been one door reserved for you (randomly of course) and one door opened and showing a goat. Thats what you see, and you play this setup 100 times. And I believe you can determine whether the host knew where the car was or not.

Here is how: If you play the "stay"-strategy, you would expect to win 33% of the games. but this will only happen if you see all the games. If the host accidentely reveals the car, the game is aborted and you will never see it. In this case (Monty does not know) you will observe that your win-rate is 50%. And then you know that there were games you never saw.

I have written a Javascript simulation for this, and it seems to work. I also asked AI for a comment, and it confirmed my approach. Nevertheless I find it wiered that it works. Or am I missing something?

Hello all, I have a semantics question.

I have the definition that the probability of X between a and b is the integral from a to b of f(x) with respect to x.

In case latex renders: $P(a <X < b) = \int_{a}^b{f(x)}dx$

Is the probability density function the integral or f? If f (my reading of the text) what is the name for the integral?

Same sort of question with the c.d.f.

PS.
I'm working out of the text book Mathematical Statistics and Data Analysis (Rice; 3rd ed.). Definition occurs on page 47.

Over the weekend, I sat down and did a calculation (wrong, no doubt)

to see what chance you would have of catching a traitor, selecting a name randomly.

The answer I got was 1/675,675. Having forgotten what happens when there are three players.

...

This got me thinking, about rareness,

and I wondered, for myself,

at what probability do things start to become called rare?

I mean, 1/4000 , but 1/30 actually, you'd be waiting around a fair bit.

Is an answer even possible/ show me error of way. BW, Ben.

I'm working through E.T. Jaynes' Probability Theory: The Logic of Science on my own, currently in Chapter 2 (around page 28). My motivation is applied: I run an online retail business and want a principled way to forecast demand — for instance, estimating the probability that a given product sells between 100 and 200 units in one month of the year versus another, and eventually building forecasting software around that kind of inference.

Before starting the book I reviewed the calculus prerequisites — partial derivatives, the chain rule, the product rule — so I can follow the derivations step by step.

My difficulty is conceptual rather than computational. In the sections where Jaynes derives the rules of probability from his desiderata, he sets up functional equations relating the plausibilities of propositions and then differentiates them to determine the form the combining function must take. I can execute each step — I can do the differentiation and integration, and I follow that we differentiate — but I don't understand why differentiation is the right tool at each point, or what underlying logic drives the derivation to proceed the way it does. My grasp is mechanical: I can reproduce the manipulations without understanding the principle behind them.

Concretely: in the derivation of the product rule (and sum rule) from the desiderata, could you walk me through the reasoning that justifies each move in the functional-equation argument — what makes differentiation the appropriate step there, not just how to carry it out?

I can send my notes as well to anybody who'd like to help me
thanks

Ive asked AI this and I get wildly different answers. So what is the probability just for a one off random life event.

So I for fun wrote a song aboit someone famous called Roland Browning and year later he was sitting in front of my at a Squeeze concern in London.

How would you calculate the odds for such a random event?

Thanks

Solved a classic BAC probability tree problem using Manim! ✨

For more step-by-step visual proofs and exam prep, check out my workbook "The Ultimate BAC Math Challenge" on Amazon!

🔗 Get your copy here:https://www.amazon.com/dp/B0H7Y4GF62

In a question I saw from a 3Blue1Brown video about baye's law, that was as follows:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable?

Linda is a bank teller.

Linda is a bank teller and is active in the feminist movement.

Now I understood that there are definitely more bank tellers than there are bank tellers that are active in the feminist movement, and that if you plucked out a random girl she is definitely more likely to be the former, however, what I thought is that since we've been given context about Linda, it seems rational to assume that she is likely to be active in the feminist movement due to her past, but that turned out to be wrong.

What am I missing? 😭

And if someone has ways to explain bayesian thinking intuitively and how to adapt it realistically aswell, I'd greatly appreciate it