Gibbs sampling is a Markov Chain Monte Carlo (MCMC)
- 2. Gibbs Sampling 30.07.2022 2/22 30.07.2022 DJSCE, Mumbai
- 3. Outline 1 Introduction 2 Gibbs Sampling : Introduction 3 Robust Modeling 4 Binary Response Regression with a Probit Link 5 Missing Data and Gibbs Sampling 6 Proper Priors and Model Selection 7 Estimating a Table of Means 8 Predicting the Success of Future Students 3/22 30.07.2022 DJSCE, Mumbai
- 4. Introduction to Gibbs Sampling What is Gibbs Sampling ? Why is it important ? 4/22 30.07.2022 DJSCE, Mumbai
- 5. Why is Gibbs Sampling Important ? Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for sampling from complex probability distributions. 5/22 30.07.2022 DJSCE, Mumbai
- 6. Why is Gibbs Sampling Important ? Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for sampling from complex probability distributions. It is particularly useful when it’s challenging to sample directly from the joint distribution of multiple variables. 5/22 30.07.2022 DJSCE, Mumbai
- 7. Why is Gibbs Sampling Important ? Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for sampling from complex probability distributions. It is particularly useful when it’s challenging to sample directly from the joint distribution of multiple variables. 5/22 30.07.2022 DJSCE, Mumbai
- 8. Why is Gibbs Sampling Important ? Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for sampling from complex probability distributions. It is particularly useful when it’s challenging to sample directly from the joint distribution of multiple variables. Gibbs Sampling allows us to explore high-dimensional spaces and estimate complex probabilistic models. 5/22 30.07.2022 DJSCE, Mumbai
- 9. Why is Gibbs Sampling Important ? Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for sampling from complex probability distributions. It is particularly useful when it’s challenging to sample directly from the joint distribution of multiple variables. Gibbs Sampling allows us to explore high-dimensional spaces and estimate complex probabilistic models. It’s widely used in Bayesian statistics, machine learning, and data analysis. 5/22 30.07.2022 DJSCE, Mumbai
- 10. Why is Gibbs Sampling Important ? Gibbs Sampling is a Markov Chain Monte Carlo (MCMC) technique used for sampling from complex probability distributions. It is particularly useful when it’s challenging to sample directly from the joint distribution of multiple variables. Gibbs Sampling allows us to explore high-dimensional spaces and estimate complex probabilistic models. It’s widely used in Bayesian statistics, machine learning, and data analysis. Gibbs Sampling is a fundamental tool for Bayesian inference, which helps us make probabilistic inferences about unknown quantities. 5/22 30.07.2022 DJSCE, Mumbai
- 11. Markov Chain Monte Carlo (MCMC) Technique MCMC is a class of algorithms used for statistical sampling and numerical integration. 6/22 30.07.2022 DJSCE, Mumbai
- 12. Markov Chain Monte Carlo (MCMC) Technique MCMC is a class of algorithms used for statistical sampling and numerical integration. It’s particularly useful for estimating complex probability distributions. 6/22 30.07.2022 DJSCE, Mumbai
- 13. Markov Chain Monte Carlo (MCMC) Technique MCMC is a class of algorithms used for statistical sampling and numerical integration. It’s particularly useful for estimating complex probability distributions. Central to MCMC methods is the concept of a Markov chain, a sequence of random variables where each depends only on the previous one. 6/22 30.07.2022 DJSCE, Mumbai
- 14. Statistical Sampling and Numerical Integration (Layman’s Terms) Imagine you have a big bag of candies. You want to know the average color of candies in the bag without checking every single one. 7/22 30.07.2022 DJSCE, Mumbai
- 15. Statistical Sampling and Numerical Integration (Layman’s Terms) Imagine you have a big bag of candies. You want to know the average color of candies in the bag without checking every single one. Statistical Sampling is like taking a handful of candies from the bag and using that small group to estimate the average color of all the candies. It’s a shortcut that gives you a good idea without checking every candy. 7/22 30.07.2022 DJSCE, Mumbai
- 16. Statistical Sampling and Numerical Integration (Layman’s Terms) Imagine you have a big bag of candies. You want to know the average color of candies in the bag without checking every single one. Statistical Sampling is like taking a handful of candies from the bag and using that small group to estimate the average color of all the candies. It’s a shortcut that gives you a good idea without checking every candy. Now, let’s talk about Numerical Integration. Imagine you want to know how much paint you need to cover a curved wall. You can’t measure it directly, so you divide the wall into many small rectangles, find the area of each rectangle, and add them up. This is a simple way to estimate the total paint needed. 7/22 30.07.2022 DJSCE, Mumbai
- 17. Statistical Sampling and Numerical Integration (Layman’s Terms) Imagine you have a big bag of candies. You want to know the average color of candies in the bag without checking every single one. Statistical Sampling is like taking a handful of candies from the bag and using that small group to estimate the average color of all the candies. It’s a shortcut that gives you a good idea without checking every candy. Now, let’s talk about Numerical Integration. Imagine you want to know how much paint you need to cover a curved wall. You can’t measure it directly, so you divide the wall into many small rectangles, find the area of each rectangle, and add them up. This is a simple way to estimate the total paint needed. Numerical Integration is similar. It’s a method to estimate complicated math problems that can’t be solved directly. You break them down into simpler pieces, approximate each piece, and add them up to get a good answer. 7/22 30.07.2022 DJSCE, Mumbai
- 18. Estimating Candy Colors with MCMC We still have our big bag of candies, and we want to know the average color of candies inside it without checking every single one. 8/22 30.07.2022 DJSCE, Mumbai
- 19. Estimating Candy Colors with MCMC We still have our big bag of candies, and we want to know the average color of candies inside it without checking every single one. Suppose we have a special machine called the ”Candy Color Analyzer” (MCMC). This machine can randomly sample a few candies from the bag and determine their color. 8/22 30.07.2022 DJSCE, Mumbai
- 20. Estimating Candy Colors with MCMC We still have our big bag of candies, and we want to know the average color of candies inside it without checking every single one. Suppose we have a special machine called the ”Candy Color Analyzer” (MCMC). This machine can randomly sample a few candies from the bag and determine their color. The Candy Color Analyzer is like a detective. It starts with a guess for the average color, and it keeps sampling candies and adjusting its guess to get closer and closer to the true average color. 8/22 30.07.2022 DJSCE, Mumbai
- 21. Estimating Candy Colors with MCMC We still have our big bag of candies, and we want to know the average color of candies inside it without checking every single one. Suppose we have a special machine called the ”Candy Color Analyzer” (MCMC). This machine can randomly sample a few candies from the bag and determine their color. The Candy Color Analyzer is like a detective. It starts with a guess for the average color, and it keeps sampling candies and adjusting its guess to get closer and closer to the true average color. The more candies the machine samples and adjusts its guess, the more accurate its estimate becomes. It’s like playing a guessing game, but it’s getting better with each turn. 8/22 30.07.2022 DJSCE, Mumbai
- 22. Estimating Candy Colors with MCMC We still have our big bag of candies, and we want to know the average color of candies inside it without checking every single one. Suppose we have a special machine called the ”Candy Color Analyzer” (MCMC). This machine can randomly sample a few candies from the bag and determine their color. The Candy Color Analyzer is like a detective. It starts with a guess for the average color, and it keeps sampling candies and adjusting its guess to get closer and closer to the true average color. The more candies the machine samples and adjusts its guess, the more accurate its estimate becomes. It’s like playing a guessing game, but it’s getting better with each turn. In the end, the Candy Color Analyzer (MCMC) provides a pretty good estimate of the average color without checking every single candy individually. 8/22 30.07.2022 DJSCE, Mumbai
- 23. Statistical Sampling and Numerical Integration (Layman’s Examples) Imagine you have a garden full of different types of flowers. You want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! 9/22 30.07.2022 DJSCE, Mumbai
- 24. Statistical Sampling and Numerical Integration (Layman’s Examples) Imagine you have a garden full of different types of flowers. You want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Statistical Sampling is like picking a few flowers from different parts of your garden and measuring their height. You use these measurements to estimate the average height of all the flowers. It’s like getting a taste of your whole garden without having to check every single flower. 9/22 30.07.2022 DJSCE, Mumbai
- 25. Statistical Sampling and Numerical Integration (Layman’s Examples) Imagine you have a garden full of different types of flowers. You want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Statistical Sampling is like picking a few flowers from different parts of your garden and measuring their height. You use these measurements to estimate the average height of all the flowers. It’s like getting a taste of your whole garden without having to check every single flower. Now, let’s talk about Numerical Integration. Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. 9/22 30.07.2022 DJSCE, Mumbai
- 26. Statistical Sampling and Numerical Integration (Layman’s Examples) Imagine you have a garden full of different types of flowers. You want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Statistical Sampling is like picking a few flowers from different parts of your garden and measuring their height. You use these measurements to estimate the average height of all the flowers. It’s like getting a taste of your whole garden without having to check every single flower. Now, let’s talk about Numerical Integration. Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. Numerical Integration is like keeping track of the number of guests at different times during the party. You add up the number of guests at each hour to estimate the total number of guests for the whole day, helping you decide how much food to prepare. 9/22 30.07.2022 DJSCE, Mumbai
- 27. Estimating Average Flower Height with MCMC Imagine you have a garden full of different types of flowers, and you want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! 10/22 30.07.2022 DJSCE, Mumbai
- 28. Estimating Average Flower Height with MCMC Imagine you have a garden full of different types of flowers, and you want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Instead of measuring every single flower, you have a magic garden ”Height Estimator” (MCMC) tool. 10/22 30.07.2022 DJSCE, Mumbai
- 29. Estimating Average Flower Height with MCMC Imagine you have a garden full of different types of flowers, and you want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Instead of measuring every single flower, you have a magic garden ”Height Estimator” (MCMC) tool. The Height Estimator starts with a guess for the average flower height. It’s like taking an educated guess based on a few flowers. 10/22 30.07.2022 DJSCE, Mumbai
- 30. Estimating Average Flower Height with MCMC Imagine you have a garden full of different types of flowers, and you want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Instead of measuring every single flower, you have a magic garden ”Height Estimator” (MCMC) tool. The Height Estimator starts with a guess for the average flower height. It’s like taking an educated guess based on a few flowers. Now, the magic happens. The Height Estimator randomly selects a few flowers, measures their height, and updates its guess for the average height. 10/22 30.07.2022 DJSCE, Mumbai
- 31. Estimating Average Flower Height with MCMC Imagine you have a garden full of different types of flowers, and you want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Instead of measuring every single flower, you have a magic garden ”Height Estimator” (MCMC) tool. The Height Estimator starts with a guess for the average flower height. It’s like taking an educated guess based on a few flowers. Now, the magic happens. The Height Estimator randomly selects a few flowers, measures their height, and updates its guess for the average height. It repeats this process, picking different flowers each time. The more flowers it measures and updates its guess, the closer it gets to the true average height. 10/22 30.07.2022 DJSCE, Mumbai
- 32. Estimating Average Flower Height with MCMC Imagine you have a garden full of different types of flowers, and you want to know the average height of all the flowers in your garden. Counting each flower is a lot of work ! Instead of measuring every single flower, you have a magic garden ”Height Estimator” (MCMC) tool. The Height Estimator starts with a guess for the average flower height. It’s like taking an educated guess based on a few flowers. Now, the magic happens. The Height Estimator randomly selects a few flowers, measures their height, and updates its guess for the average height. It repeats this process, picking different flowers each time. The more flowers it measures and updates its guess, the closer it gets to the true average height. Eventually, the Height Estimator gives you a very good estimate of the average flower height in your garden without having to measure every single flower. 10/22 30.07.2022 DJSCE, Mumbai
- 33. Estimating Food for a Fluctuating Party with MCMC Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. 11/22 30.07.2022 DJSCE, Mumbai
- 34. Estimating Food for a Fluctuating Party with MCMC Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. It’s like trying to figure out how many slices of pizza to order when you don’t know how many friends will drop by. 11/22 30.07.2022 DJSCE, Mumbai
- 35. Estimating Food for a Fluctuating Party with MCMC Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. It’s like trying to figure out how many slices of pizza to order when you don’t know how many friends will drop by. Imagine you have a special ”Party Guest Counter” (MCMC) tool. This tool helps you estimate the number of guests at any given time. 11/22 30.07.2022 DJSCE, Mumbai
- 36. Estimating Food for a Fluctuating Party with MCMC Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. It’s like trying to figure out how many slices of pizza to order when you don’t know how many friends will drop by. Imagine you have a special ”Party Guest Counter” (MCMC) tool. This tool helps you estimate the number of guests at any given time. The Party Guest Counter starts with an initial estimate of guests, and it’s continuously updating its guess as people arrive and leave. 11/22 30.07.2022 DJSCE, Mumbai
- 37. Estimating Food for a Fluctuating Party with MCMC Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. It’s like trying to figure out how many slices of pizza to order when you don’t know how many friends will drop by. Imagine you have a special ”Party Guest Counter” (MCMC) tool. This tool helps you estimate the number of guests at any given time. The Party Guest Counter starts with an initial estimate of guests, and it’s continuously updating its guess as people arrive and leave. It’s like a party detective – every so often, it takes a quick headcount, adjusts its guess, and becomes better at estimating the number of guests. 11/22 30.07.2022 DJSCE, Mumbai
- 38. Estimating Food for a Fluctuating Party with MCMC Suppose you’re throwing a big party, and you want to know how much food to buy. The number of guests keeps changing throughout the day. It’s like trying to figure out how many slices of pizza to order when you don’t know how many friends will drop by. Imagine you have a special ”Party Guest Counter” (MCMC) tool. This tool helps you estimate the number of guests at any given time. The Party Guest Counter starts with an initial estimate of guests, and it’s continuously updating its guess as people arrive and leave. It’s like a party detective – every so often, it takes a quick headcount, adjusts its guess, and becomes better at estimating the number of guests. As the party progresses, the Party Guest Counter (MCMC) provides you with a good estimate of the number of guests, helping you order just the right amount of pizza and snacks, even if the guest list keeps changing. 11/22 30.07.2022 DJSCE, Mumbai
- 39. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. 12/22 30.07.2022 DJSCE, Mumbai
- 40. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. In many real-world scenarios, such as Bayesian modeling, it’s common to encounter these complex distributions. 12/22 30.07.2022 DJSCE, Mumbai
- 41. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. In many real-world scenarios, such as Bayesian modeling, it’s common to encounter these complex distributions. Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are valuable tools for sampling from such distributions. 12/22 30.07.2022 DJSCE, Mumbai
- 42. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. In many real-world scenarios, such as Bayesian modeling, it’s common to encounter these complex distributions. Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are valuable tools for sampling from such distributions. 12/22 30.07.2022 DJSCE, Mumbai
- 43. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. In many real-world scenarios, such as Bayesian modeling, it’s common to encounter these complex distributions. Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are valuable tools for sampling from such distributions. Multivariate Normal Distributions : When dealing with multiple correlated variables, the joint distribution can become complex. Gibbs Sampling can be used for sampling from multivariate normal distributions. 12/22 30.07.2022 DJSCE, Mumbai
- 44. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. In many real-world scenarios, such as Bayesian modeling, it’s common to encounter these complex distributions. Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are valuable tools for sampling from such distributions. Multivariate Normal Distributions : When dealing with multiple correlated variables, the joint distribution can become complex. Gibbs Sampling can be used for sampling from multivariate normal distributions. Bayesian Hierarchical Models : Hierarchical models with multiple layers of parameters can lead to complex posterior distributions. Gibbs Sampling simplifies sampling in such cases. 12/22 30.07.2022 DJSCE, Mumbai
- 45. Examples of Complex Probability Distributions Complex probability distributions are those that cannot be easily expressed using standard probability density functions. In many real-world scenarios, such as Bayesian modeling, it’s common to encounter these complex distributions. Markov Chain Monte Carlo (MCMC) methods like Gibbs Sampling are valuable tools for sampling from such distributions. Multivariate Normal Distributions : When dealing with multiple correlated variables, the joint distribution can become complex. Gibbs Sampling can be used for sampling from multivariate normal distributions. Bayesian Hierarchical Models : Hierarchical models with multiple layers of parameters can lead to complex posterior distributions. Gibbs Sampling simplifies sampling in such cases. Non-conjugate Priors : When the prior and likelihood are not conjugate, the posterior distribution may be difficult to compute analytically. Gibbs Sampling provides a solution. 12/22 30.07.2022 DJSCE, Mumbai
- 46. Complex Probability Distributions (In Everyday Terms) We’ve seen how to estimate things like candy colors, flower heights, and party guests using tools like MCMC. But what if the problem becomes even more challenging ? 13/22 30.07.2022 DJSCE, Mumbai
- 47. Complex Probability Distributions (In Everyday Terms) We’ve seen how to estimate things like candy colors, flower heights, and party guests using tools like MCMC. But what if the problem becomes even more challenging ? Complex Probability Distributions are like really tricky problems where you can’t simply use basic tools. They’re like puzzles that are hard to solve. 13/22 30.07.2022 DJSCE, Mumbai
- 48. Complex Probability Distributions (In Everyday Terms) We’ve seen how to estimate things like candy colors, flower heights, and party guests using tools like MCMC. But what if the problem becomes even more challenging ? Complex Probability Distributions are like really tricky problems where you can’t simply use basic tools. They’re like puzzles that are hard to solve. For example, think about estimating the probability of finding a specific rare flower in your garden based on its color, size, and location. That’s a complex probability distribution problem. 13/22 30.07.2022 DJSCE, Mumbai
- 49. Complex Probability Distributions (In Everyday Terms) We’ve seen how to estimate things like candy colors, flower heights, and party guests using tools like MCMC. But what if the problem becomes even more challenging ? Complex Probability Distributions are like really tricky problems where you can’t simply use basic tools. They’re like puzzles that are hard to solve. For example, think about estimating the probability of finding a specific rare flower in your garden based on its color, size, and location. That’s a complex probability distribution problem. Complex probability distributions are like the most challenging party guest scenarios. Not only do guests come and go, but their behavior and preferences change too. It’s hard to predict and manage. 13/22 30.07.2022 DJSCE, Mumbai
- 50. Complex Probability Distributions (In Everyday Terms) We’ve seen how to estimate things like candy colors, flower heights, and party guests using tools like MCMC. But what if the problem becomes even more challenging ? Complex Probability Distributions are like really tricky problems where you can’t simply use basic tools. They’re like puzzles that are hard to solve. For example, think about estimating the probability of finding a specific rare flower in your garden based on its color, size, and location. That’s a complex probability distribution problem. Complex probability distributions are like the most challenging party guest scenarios. Not only do guests come and go, but their behavior and preferences change too. It’s hard to predict and manage. When we encounter complex probability distributions, MCMC comes to the rescue. It’s like having a magical problem solver for the most complicated puzzles and unpredictable parties. 13/22 30.07.2022 DJSCE, Mumbai
- 51. Complex Probability Distributions with a Coin Toss Let’s dive into a numerical example. Imagine we have a biased coin. 14/22 30.07.2022 DJSCE, Mumbai
- 52. Complex Probability Distributions with a Coin Toss Let’s dive into a numerical example. Imagine we have a biased coin. The probability of getting a head (H) is 0.3, and the probability of getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin. 14/22 30.07.2022 DJSCE, Mumbai
- 53. Complex Probability Distributions with a Coin Toss Let’s dive into a numerical example. Imagine we have a biased coin. The probability of getting a head (H) is 0.3, and the probability of getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin. Now, let’s say we want to know the probability of getting two heads (HH) in a row when we flip this coin twice. 14/22 30.07.2022 DJSCE, Mumbai
- 54. Complex Probability Distributions with a Coin Toss Let’s dive into a numerical example. Imagine we have a biased coin. The probability of getting a head (H) is 0.3, and the probability of getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin. Now, let’s say we want to know the probability of getting two heads (HH) in a row when we flip this coin twice. To calculate this, we need to deal with complex probability distributions. It’s not just straightforward like flipping a fair coin. 14/22 30.07.2022 DJSCE, Mumbai
- 55. Complex Probability Distributions with a Coin Toss Let’s dive into a numerical example. Imagine we have a biased coin. The probability of getting a head (H) is 0.3, and the probability of getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin. Now, let’s say we want to know the probability of getting two heads (HH) in a row when we flip this coin twice. To calculate this, we need to deal with complex probability distributions. It’s not just straightforward like flipping a fair coin. Here’s where MCMC comes in. It can simulate thousands of coin flips, accounting for the biased probabilities, and estimate the probability of getting HH. 14/22 30.07.2022 DJSCE, Mumbai
- 56. Complex Probability Distributions with a Coin Toss Let’s dive into a numerical example. Imagine we have a biased coin. The probability of getting a head (H) is 0.3, and the probability of getting a tail (T) is 0.7. This coin doesn’t behave like a fair coin. Now, let’s say we want to know the probability of getting two heads (HH) in a row when we flip this coin twice. To calculate this, we need to deal with complex probability distributions. It’s not just straightforward like flipping a fair coin. Here’s where MCMC comes in. It can simulate thousands of coin flips, accounting for the biased probabilities, and estimate the probability of getting HH. Using MCMC, we might find that the probability of HH is approximately 0.09 for this biased coin. 14/22 30.07.2022 DJSCE, Mumbai
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- 58. Complex Probability Distributions with Playing Cards Consider a standard deck of playing cards with 52 cards. Now, let’s say you want to know the probability of drawing two red cards (hearts or diamonds) in a row without replacement. 15/22 30.07.2022 DJSCE, Mumbai
- 59. Complex Probability Distributions with Playing Cards Consider a standard deck of playing cards with 52 cards. Now, let’s say you want to know the probability of drawing two red cards (hearts or diamonds) in a row without replacement. To calculate this probability, we need to deal with complex probability distributions. Unlike flipping a fair coin, it’s not straightforward. 15/22 30.07.2022 DJSCE, Mumbai
- 60. Complex Probability Distributions with Playing Cards Consider a standard deck of playing cards with 52 cards. Now, let’s say you want to know the probability of drawing two red cards (hearts or diamonds) in a row without replacement. To calculate this probability, we need to deal with complex probability distributions. Unlike flipping a fair coin, it’s not straightforward. In this case, the probability of drawing a red card on the first draw is 26/52, and on the second draw (without replacement), it becomes 25/51. 15/22 30.07.2022 DJSCE, Mumbai
- 61. Complex Probability Distributions with Playing Cards Consider a standard deck of playing cards with 52 cards. Now, let’s say you want to know the probability of drawing two red cards (hearts or diamonds) in a row without replacement. To calculate this probability, we need to deal with complex probability distributions. Unlike flipping a fair coin, it’s not straightforward. In this case, the probability of drawing a red card on the first draw is 26/52, and on the second draw (without replacement), it becomes 25/51. Complex probability distributions arise when we want to find the probability of drawing two specific cards in a row from a deck with different suits and ranks. 15/22 30.07.2022 DJSCE, Mumbai
- 62. Complex Probability Distributions with Playing Cards Consider a standard deck of playing cards with 52 cards. Now, let’s say you want to know the probability of drawing two red cards (hearts or diamonds) in a row without replacement. To calculate this probability, we need to deal with complex probability distributions. Unlike flipping a fair coin, it’s not straightforward. In this case, the probability of drawing a red card on the first draw is 26/52, and on the second draw (without replacement), it becomes 25/51. Complex probability distributions arise when we want to find the probability of drawing two specific cards in a row from a deck with different suits and ranks. This is where MCMC comes to the rescue. It can simulate thousands of card draws, account for the changing probabilities with each draw, and estimate the probability of drawing two red cards in a row. 15/22 30.07.2022 DJSCE, Mumbai
- 63. Complex Probability Distributions with Playing Cards Consider a standard deck of playing cards with 52 cards. Now, let’s say you want to know the probability of drawing two red cards (hearts or diamonds) in a row without replacement. To calculate this probability, we need to deal with complex probability distributions. Unlike flipping a fair coin, it’s not straightforward. In this case, the probability of drawing a red card on the first draw is 26/52, and on the second draw (without replacement), it becomes 25/51. Complex probability distributions arise when we want to find the probability of drawing two specific cards in a row from a deck with different suits and ranks. This is where MCMC comes to the rescue. It can simulate thousands of card draws, account for the changing probabilities with each draw, and estimate the probability of drawing two red cards in a row. Using MCMC, we might find that the probability of drawing two red cards in a row is approximately 0.244. 15/22 30.07.2022 DJSCE, Mumbai
- 64. Robust Modeling Discuss the concept of robust modeling. How is Gibbs Sampling used in robust modeling ? 16/22 30.07.2022 DJSCE, Mumbai
- 65. Binary Response Regression with a Probit Link Explain binary response regression with a probit link. Show practical applications and examples. 17/22 30.07.2022 DJSCE, Mumbai
- 66. Missing Data and Gibbs Sampling How does Gibbs Sampling handle missing data ? Provide examples of its application. 18/22 30.07.2022 DJSCE, Mumbai
- 67. Proper Priors and Model Selection Explain the importance of proper priors in Gibbs Sampling. Discuss how Gibbs Sampling aids in model selection. 19/22 30.07.2022 DJSCE, Mumbai
- 68. Estimating a Table of Means Show how Gibbs Sampling can be used to estimate a table of means. Compare a flat prior over the restricted space with a hierarchical regression prior. 20/22 30.07.2022 DJSCE, Mumbai
- 69. Predicting the Success of Future Students Discuss the application of Gibbs Sampling in predicting the success of future students. Share the significance of this application. 21/22 30.07.2022 DJSCE, Mumbai
- 70. Conclusion Summarize the key points. Highlight the importance of Gibbs Sampling in various domains. 22/22 30.07.2022 DJSCE, Mumbai