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round, 22 coins yielded a head and y2 coins yielded a tail. Once again, 22 + 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(a)+log P(a₂)++log P(am). Maximizing this quantity is equivalent to minimizing the -log P(A) = -log P(as) - log P(az) - log P(am). 3. Here you need to find out P(a,) for yourself. ---

round, 22 coins yielded a head and y2 coins yielded a tail. Once again, 22 + 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(a)+log P(a₂)++log P(am). Maximizing this quantity is equivalent to minimizing the -log P(A) = -log P(as) - log P(az) - log P(am). 3. Here you need to find out P(a,) for yourself. ---

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 54E
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8:44 PM Fri Jul 28 1 of 2 1st to 2nd to 3rd to 4th to 5th to 6th to 7th toss T H mn ⠀⠀ H July 7, 2023 H T HTT HTT T HHT THT H TTTTTH H T T T T 77172 log P(A) T Now, define q = . Then the equation becomes: I H T TTH H T T 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 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: H log P(A) = -Σ" \ogp - Στ, " log (1 – p) = -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? VPN 100% I
Transcribed Image Text:8:44 PM Fri Jul 28 1 of 2 1st to 2nd to 3rd to 4th to 5th to 6th to 7th toss T H mn ⠀⠀ H July 7, 2023 H T HTT HTT T HHT THT H TTTTTH H T T T T 77172 log P(A) T Now, define q = . Then the equation becomes: I H T TTH H T T 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 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: H log P(A) = -Σ" \ogp - Στ, " log (1 – p) = -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? VPN 100% I
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