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round, 2 coins yielded a head and y2 coins yielded a tail. Once again, x2 + 3/2 = n. She does this experiment m times. Your job is to estimate the probability p of a coin yielding a head. 1. What is your guess on the value of p? 2. In Maximum Likelihood Estimation, we want to find a parameter p which maximizes all the observations in the dataset. If the dataset is a matrix A, where each row a1, a2,, am are individual observations, we want to maximize P(A) = P(a₁) P(az) P(am) because individ- ual experiments are independent. Maximizing this is equivalent to maximizing log P(A) = log P(a1) +log P(a2)++log P(am). Maximizing this quantity is equivalent to minimizing the -log P(A) = -log P(as) - log P(a2) --log P(am). 3. Here you need to find out P(a) for yourself. 4. If you can do that properly, you will find an equation of the form: . Σ., M: log (1 - 1) mn Now, define 9 log P(A) mn Sm 21 mn logp- Then the equation becomes: log P(A) mn -q logp - (1-q) log (1 - p) Use Pinsker's Inequality or Calculus to show that, p = q. 5. What is the value of p for the above dataset given in the table? 6. If you toss 20 coins now, how many coins are most likely to yield a head?

round, 2 coins yielded a head and y2 coins yielded a tail. Once again, x2 + 3/2 = n. She does this experiment m times. Your job is to estimate the probability p of a coin yielding a head. 1. What is your guess on the value of p? 2. In Maximum Likelihood Estimation, we want to find a parameter p which maximizes all the observations in the dataset. If the dataset is a matrix A, where each row a1, a2,, am are individual observations, we want to maximize P(A) = P(a₁) P(az) P(am) because individ- ual experiments are independent. Maximizing this is equivalent to maximizing log P(A) = log P(a1) +log P(a2)++log P(am). Maximizing this quantity is equivalent to minimizing the -log P(A) = -log P(as) - log P(a2) --log P(am). 3. Here you need to find out P(a) for yourself. 4. If you can do that properly, you will find an equation of the form: . Σ., M: log (1 - 1) mn Now, define 9 log P(A) mn Sm 21 mn logp- Then the equation becomes: log P(A) mn -q logp - (1-q) log (1 - p) Use Pinsker's Inequality or Calculus to show that, p = q. 5. What is the value of p for the above dataset given in the table? 6. If you toss 20 coins now, how many coins are most likely to yield a head?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1st to 2nd to 3rd to 4th to 5th to 6th to 7th toss TT HTTH Now, define q = Dm H mn HTTHT T T T H T TTH TTT H TTT T T H H Η _log P(A) = mn T H Then the equation becomes: log P(A) mn Machine Learning is the science of learning from experience. Suppose Alice is repeatedly doing an experiment. In each experiment she tosses n coins. She does this experiment m times. In the first round, ₁ coins yielded a head and y₁ coins yielded a tail. Notice that, z₁+y₁ = n. In the second T H round, 2 coins yielded a head and y2 coins yielded a tail. Once again, x2 + y2 = n. She does this experiment m times. Your job is to estimate the probability p of a coin yielding a head. 1. What is your guess on the value of p? 2. In Maximum Likelihood Estimation, we want to find a parameter p which maximizes all the observations in the dataset. If the dataset is a matrix A, where each row a1, a2,,am are individual observations, we want to maximize P(A) = P(a₁)P(a₂) P(am) because individ- ual experiments are independent. Maximizing this is equivalent to maximizing log P(A) = log P(a₁) +log P(a₂)++log P(am). Maximizing this quantity is equivalent to minimizing the -log P(A) = -log P(a1) - log P(a2)log P(am). 3. Here you need to find out P(a;) for yourself. 4. If you can do that properly, you will find an equation of the form: logp-log (1-p) H T H H T T =-q logp (1-q) log (1 - p) Use Pinsker's Inequality or Calculus to show that, p = q. 5. What is the value of p for the above dataset given in the table? 6. If you toss 20 coins now, how many coins are most likely to yield a head?
Transcribed Image Text:1st to 2nd to 3rd to 4th to 5th to 6th to 7th toss TT HTTH Now, define q = Dm H mn HTTHT T T T H T TTH TTT H TTT T T H H Η _log P(A) = mn T H Then the equation becomes: log P(A) mn Machine Learning is the science of learning from experience. Suppose Alice is repeatedly doing an experiment. In each experiment she tosses n coins. She does this experiment m times. In the first round, ₁ coins yielded a head and y₁ coins yielded a tail. Notice that, z₁+y₁ = n. In the second T H round, 2 coins yielded a head and y2 coins yielded a tail. Once again, x2 + y2 = n. She does this experiment m times. Your job is to estimate the probability p of a coin yielding a head. 1. What is your guess on the value of p? 2. In Maximum Likelihood Estimation, we want to find a parameter p which maximizes all the observations in the dataset. If the dataset is a matrix A, where each row a1, a2,,am are individual observations, we want to maximize P(A) = P(a₁)P(a₂) P(am) because individ- ual experiments are independent. Maximizing this is equivalent to maximizing log P(A) = log P(a₁) +log P(a₂)++log P(am). Maximizing this quantity is equivalent to minimizing the -log P(A) = -log P(a1) - log P(a2)log P(am). 3. Here you need to find out P(a;) for yourself. 4. If you can do that properly, you will find an equation of the form: logp-log (1-p) H T H H T T =-q logp (1-q) log (1 - p) Use Pinsker's Inequality or Calculus to show that, p = q. 5. What is the value of p for the above dataset given in the table? 6. If you toss 20 coins now, how many coins are most likely to yield a head?
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