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Thank you for this absolute gem of a lecture. I think it might tickle you to know how I approached the problem re Tanya's tennis and sunny days (23:00 ish). As a healthcare professional, I'm much more familiar with sensitivity, specificity, positive/negative predictive values, prevalence etc, than I am with an equation for conditional probability. So for my first attempt at the problem, I basically framed the situation as "tennis" being a diagnostic test for "sunny", and drew a 2x2 table for "tennis" against "sunny"! The sensitivity of the test is 80%, ie. 80% of "sunnies" had a positive "tennis" result. The specificity of the test is 65%, ie. 65% of "not sunnies" had a negative "tennis" result. The "prevalence" of sunny is 60% and not sunny is 40%. Therefore, I just had to solve for the positive predictive value of the "tennis" test for "sunny", by using the relative prevalence to "weight" the sunnies vs not-sunnies within the "tennis" group. Et voila, it yielded the exact same process of multiplication as using the cond probability equation, which I used for my second attempt. I know this may sound like a much more complicated method, but seeing the probabilities in a table and applying concepts I already know truly helped me actually understand the multiplications within the conditional probability equation, rather than just solving for it blindly. This was a lightbulb moment for me. Thank you!

thanks dawg

This was an awesome video; really appreciate it!

This video deserves more likes.

I just love the flow of the lecture.

thank you

Really appreciate your explanation of Q13 - I wondered how the mark scheme started with (2a-b)^2≥0

Excellent. In general I leave comments 0.5% of the time. But when I think something is really superb I always leave a comment. What is the chance I thought Woody’s tutorial was really superb?

Timestamps would help greatly. Great course anyways ❤

Question: at 34:18 The question is if a child take the pill then how likely they will be a genius. Why at 35:25, it is that being a genius is a given? shouldn't it be P(genius | pill)?

🎯 Key Takeaways for quick navigation: 00:00 *🎓 Introduction to Bayesian Statistics* - Exploring Bayesian statistics from scratch. - Suitable for anyone interested in probability and statistics, from students to professionals. - Starting with fundamental questions about probability and its applications. 01:10 *🎲 Objective vs. Subjective Views on Probability* - Contrasting objective (frequentist) and subjective (Bayesian) views on probability. - Highlighting limitations of frequentist approach, especially for one-off events like horse races. - Illustrating subjective Bayesian model's flexibility and rationality in handling uncertainty. 09:33 *📊 Degrees of Belief in Bayesian Probability* - Bayesian probability as degrees of belief or uncertainty measures. - Illustrating subjective probabilities through scenarios involving pregnancy and gender prediction. - Emphasizing rationality in adjusting beliefs based on available evidence. 10:01 *🧠 Conditional Probability Basics* - Introduction to conditional probability using simple visual examples. - Building intuition for conditional probability through visualizations. - Setting the stage for understanding Bayes' theorem. 13:19 *📝 Formulating Baby Bayes Theorem* - Deriving a simplified version of Bayes' theorem using visual probability representations. - Demonstrating application of the theorem in simple probability problems. - Introducing notation and terminology for hypothesis and evidence probabilities. 20:20 *🌳 Bayes Theorem Application with Tree Diagrams* - Applying Bayes' theorem to complex scenarios using tree diagrams. - Solving probability problems involving multiple events and conditional probabilities. - Demonstrating how evidence updates prior probabilities to yield posterior probabilities. 23:34 *📈 Bayesian statistics application example: Updating probability with evidence* - Bayes' theorem updates the probability of an event given new evidence. - Example: Given the probability of sunny weather and playing tennis, Bayes' theorem helps update the probability of sunny weather given that tennis was played. - Demonstrates how prior beliefs are adjusted based on new information. 24:45 *📊 Bayesian statistics application example: Probabilistic analysis in economics* - Scenario: Analyzing the probability of a recession given job loss using Bayes' theorem. - Demonstrates the use of prior probabilities and conditional probabilities in economic analysis. - Shows how Bayesian statistics can be applied to decision-making in economic forecasting. 29:22 *🏃‍♀️ Bayesian statistics application example: Probability distributions in sports* - Example: Analyzing the probability of a girl running 100 meters in a certain time frame using normal distribution. - Shows how Bayesian statistics is used to update probabilities based on additional information (e.g., being in the school running team). - Illustrates how conditional probability influences the assessment of outcomes in sports. 33:19 *🧠 Bayesian statistics application example: Counter-intuitive results* - Examines counter-intuitive outcomes using conditional probability in IQ distribution scenarios. - Demonstrates how small changes in distributions can lead to significant shifts in probabilities. - Highlights the importance of understanding conditional probability in interpreting statistical results. 41:42 *🦠 Bayesian statistics application example: Medical diagnosis* - Examines a medical diagnosis scenario using Bayes' theorem. - Illustrates how prior beliefs are updated based on diagnostic test results. - Emphasizes the significance of understanding conditional probability in medical decision-making. 48:11 *📊 Understanding Bayes Theorem through an Example* - Explains the application of Bayes Theorem using an example involving Steve, a shy individual, to illustrate how prior probabilities and evidence combine. - Demonstrates how intuition can be misleading when prior probabilities and evidence are not considered. - Breaks down the calculation process step by step, showing the application of Bayes Theorem in determining the probability of Steve being a librarian given certain traits. 51:42 *📈 Formal Naming and Components of Bayes Theorem* - Defines the formal components of Bayes Theorem: prior, posterior, likelihood, and evidence. - Illustrates the terminology used in relation to each component, such as "prior" for the initial probability, "posterior" for the updated probability, "likelihood" for the probability of evidence given a hypothesis, and "evidence" for the total probability of the observed evidence. - Provides insights into the significance of each component in Bayesian inference and decision-making processes. 56:36 *🔍 Exploring a Complex Example: Bayesian Approach vs. Frequentist Approach* - Introduces a more complex example involving a game between Alice and Bob to compare Bayesian and frequentist approaches. - Contrasts the frequentist method, which relies on straightforward calculations, with the Bayesian method, which involves applying Bayes Theorem to update probabilities based on evidence. - Demonstrates how Bayesian inference can provide more accurate predictions by considering prior probabilities and updating them with observed evidence, even in complex scenarios. 48:40 *📊 Bayesian Intuition: Steve's Occupation* - Daniel Kahneman presents a scenario about Steve, a shy and tidy individual, posing the question of whether he's more likely to be a farmer or a librarian. - Despite intuitive judgment favoring Steve being a librarian, Bayesian analysis challenges this assumption by considering the proportion of farmers and librarians meeting Steve's criteria. - Applying Bayesian theorem, the analysis shows that Steve is more likely to be a farmer, emphasizing the importance of considering the base rate in making probability assessments. 51:28 *📈 Bayesian Terminology: Understanding Bayes Theorem Components* - Prior: The probability of a hypothesis before considering any new evidence. - Posterior: The probability of a hypothesis after considering new evidence. - Likelihood: The probability of observing the evidence given that the hypothesis is true. - Evidence (Marginal Likelihood): The probability of observing the evidence, accounting for both scenarios where the hypothesis is true and where it's not. 56:36 *🎲 Bayesian Approach: Alice and Bob's Game* - Illustration of a game between Alice and Bob where a ball is randomly placed on a table, dividing it into two sections, with each player scoring points based on where the ball lands. - Contrasting frequentist and Bayesian approaches in assessing Bob's probability of winning the game. - Through simulation, Bayesian analysis consistently yields a higher probability of Bob winning compared to the frequentist approach, demonstrating the Bayesian method's reliability in probabilistic assessments. Made with HARPA AI

Thanks

41:28 similar to IQ distributions for men and woman

@ 38 minutes you say "1 in a Million" and represent it as 0.00000011. "That's 1.1 in 10 million" You are getting this from a normal distribution and I think you have misinterpreted the statistic, rather than used the wrong figure. 0.1 is "1 in 10", 0.01 "1 in 100", 0.001 "1 in 1,000" etc

Thank You. So helpful!

Do you also have videos on binomial probability? Or perhaps you know of a good introductory course or book?

But aren't you assuming that P(Bob wins) is 100%? So you start with a very strong prior. Or where I am wrong? ;-)

Thank you for this video. What I missed is how you calculated the probability in your first example for normal distribution. You just said look at the area under the curve and you said calculate and get the value.

Excellent presentation. Wouldn't the Monty Hall Problem be an example of where using Bayes would be helpful? The update info would be that the host, Monty Hall will switch to a door with a goat. The setup is this: Monty Hall is the host of a TV Show, where a contestant must choose one of three doors , where there are goats behind two of the doors, and a car behind the other door. If the contestant chooses the door with the car, they get to keep it. If they choose a door with a goat behind it, they're out. The additional info is that once the contestant selects one of the doors, Monty will stop the show, open up one of the doors containing a goat, and proceed to ask the contestant if they'd prefer to switch to another door. Then the question is whether it is a good idea for the contestant to switch. Answer is yes, given by choosing at random, uniformly, the contestant will have initially chosen the car only 1/3 of the time, and one of the two goats 2/3 of the time. So, 2/3 of the time, contestant will have made the wrong choice, and will improve the odds by switching. I hope this isn't too confusing.

Thank you!

Thanks for the interesting and clear video! I have a doubt about Amira and Jane problem: why do you assume that Jane being late has an impact on Amira being late? From what we know, one could come from the Moon and the other from Mars. Did I miss something?

Great !!!

Great session sir! Just one que: Why did you take rand() < a particular column.. why not >?

Excellent video! I needed to refresh my Bayesian statistics knowledge and this was a perfect start.

Indeed your video is FANTASTIC and IMMENSILY helpful. Thanks!!!

Woah! Quality stuff and your examples added more to grasp the intuition underlying these Bayesian concepts. Regards from Pakistan <3

Correction @26:49 'we therefore know that there's a 95% chance that he will NOT lose his job ' instead of 'we therefore know that there's a 95% chance that he will lose his job '

Great lesson! Thank you. Keep up the good work.

I'm not sure if I agree with your solution at 20:13. This works assuming conditional probability (or Jane being late is influenced by Amira). But if these were independent events then the answer would just be 20%. If you know nothing about Jane and Amira, is it rational to assume it's conditional or independent? Lastly does causal reasoning play a part here? Thanks :)

“Ignore biology and assume everyone is male or female”? Huh?

👏

Brilliant course, thank you for your efforts & the material!

I knew I recognised your name! I do believe I taught you maths and/or further maths a million or so years ago? You're teaching now?!

Hi Woody, Thanks for this lesson! It is very useful. It's quite challenging to get rid of the frequentist mind after spending my entire life as so, though. I just a have a question: in the simulation, why did you calculated the last probability by using the number of rounds won by either Alice or Bob and not by the number of rounds (10000)? That's how simulations usually work. So I'll be more than grateful if you could help me out here with this doubt.

great explanation, i like how you gradually introduced the different concepts from probability -> conditional probability -> baby bayes -> bayes

This is the best explanation I ever saw.

The first lesson is everything! I finally understand the fundamental difference between Bayesian and frequentist statistics. 🎉 beautifully explained, thank you!

As I wrote before this is the best ever explanation of the Bayesian statistics - THANK YOU VERY MUCH!!! and I am coming back to it every time to refresh this concept and vocabualry :)! When we talk about the numerator ( P(a=5, b=3 | Bob wins)*P(Bob wins) ) of the probability equation (that Bob wins given A=5 and B=3) at 1:07:20. Basically we can not really operate with these terms separately (P(a=5, b=3 | Bob wins) * 0.057) and have to merge them together into the distribution (y=(8x3)x^6(1-x)^5). Just trying to catch a moment and a point when it pivots from the frequentist to the bayesian :). In other words the point is that when we look at this formula as a math expression then we can camcel out the y=(8x3)x^3(1-x)^5 while left only with (3/8)^3 which then would be purely frequentist estimation of the probability of Bob winning (I do understand that in fact we have the areas under the graphs for the respective distributions in numerator and denominator). If you were so patient to having read till this point :) - what is confusing is the P(a=5, b=3 | Bob wins) part of the numerator which is hard to imagine...

really enjoyed

excellent lecture

The best tutorial on youtube that explains Bayesian Statistics so far! 🌷

Wonderful teaching, earlier i found it difficult to understand the probability but now its seems easy. thanks prof.

For question 11 could you have used 6 then squared it to 36. If you go up by 6 you're also going out of the 4th quadrant, why would it only be horizontal and not vertical

Thank u sooo much. I really understood everything

Great video, thank you. I feel like the pill sample calculation is not correct. If there is a very tiny change in mean like 100 -> 100.0000000001, the calc made(from the video) will yield almost 50% that the genius has taken the pill which seems incorrect. I feel like it should be [P(102)-P(100)]/P(100) - I may be wrong but the original calc seems wrong to me. Also the number of zeros for calculated probability should be 5 instead of 6 zeros. Thanks

So clear, excelllent video! Was very helpful 👏🏼👏🏼 Thank you!

How can I simulate the billiards example in R?

Best explanation! Finally I get it😂

I have been interested in Bayesian analysis for a few years and seen dozens of videos. This is the best video I have seen to learn the concepts. Thank you so much for producing and sharing this knowledge!

Wow, thank you.