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Maximum Likelihood Estimation Navneet Goyal BITS, Pilani

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Maximum Likelihood Estimation (MLE) Parameter estimation is a fundamental problem in Data Analytics MLE finds the most likely value for the parameter based on the data set collected Other methods: least squares method of moments Bayesian estimation Applications of MLE Relationship with other methods

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Maximum Likelihood Estimation (MLE) The method of maximum likelihood provides estimators that have both a reasonable intuitive basis and many desirable statistical properties. The method is very broadly applicable and is simple to apply A disadvantage of the method is that it frequently requires strong assumptions about the structure of the data © 2010 by John Fox York SPIDA

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Maximum Likelihood Estimation (MLE) A coin is flipped What is the probability that it will fall Heads up? What will you do? Toss the coin a few times 3 H & 2 T Prob. is 3/5 Why? Because… Reference: Aarti Singh slides for ML Course, ML Dept. @ CMU

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Maximum Likelihood Estimation (MLE) Data D = {Xi} i=1,2,…N & Xi {H,T} P (H)= , P(T)=1- Flips are iid Reference: Aarti Singh slides for ML Course, ML Dept. @ CMU

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Population of Size of Seals blue=0 seals, red=1 seal, purple=2 seals

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Likelihood is not Probablity!!

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MLE Example We want to estimate the probability π of getting a head upon flipping a particular coin. We flip the coin ‘independently’ 10 times (i.e., we sample n = 10 flips), obtaining the following result: HHTHHHTTHH The probability of obtaining this sequence — in advance of collecting the data — is a function of the unknown parameter π : Pr(data|parameter) = Pr(HHTHHHTTHH| π ) = π 7 (1 − π ) 3 But the data for our particular sample are fixed: We have already collected them. The parameter π also has a fixed value, but this value is unknown and so we can let it vary in our imagination between 0 and 1, treating the probability of the observed data as a function of π. © 2010 by John Fox York SPIDA

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MLE Example This function is called the likelihood function: L(parameter|data) = Pr( π | HHTHHHTTHH) = π 7 (1 − π ) 3 The probability function and the likelihood function are given by the same equation, but the probability function is a function of the data with the value of the parameter fixed, while the likelihood function is a function of the parameter with the data fixed © 2010 by John Fox York SPIDA

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MLE Example Here are some representative values of the likelihood for different values of π : π L( π |data) = π 7 (1 − π ) 3 0.0 0.1 0.0000000729 0.2 0.00000655 0.3 0.0000750 0.4 0.000354 0.5 0.000977 0.6 0.00179 0.7 0.00222 0.8 0.00168 0.9 0.000478 1.0 0.0 © 2010 by John Fox York SPIDA

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MLE Example © 2010 by John Fox York SPIDA Likelihood Finction

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MLE Example © 2010 by John Fox York SPIDA Although each value of L( π |data) is a notional probability The function L( π |data) is not a probability or density function — it does not enclose an area of 1. The probability of obtaining the sample of data that we have in hand, HHTHHHTTHH is small regardless of the true value of π. This is usually the case: Any specific sample result — including the one that is realized — will have low probability. Nevertheless, the likelihood contains useful information about the unknown parameter π. For example, π cannot be 0 or 1, and is ‘unlikely’ to be close to 0 or 1. Reversing this reasoning, the value of π that is most supported by the data is the one for which the likelihood is largest. This value is the maximum-likelihood estimate (MLE), denoted by Here, =0.7, which is the sample proportion of heads, 7/10

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Likelihood Fn. (LF) Vs. PDF © 2010 by John Fox York SPIDA LF is a function of the parameter with data fixed PDF is a function of the data with parameter fixed LF is “unnormalized” probability Although each value of L( π |data) is a notional probability The function L( π |data) is not a probability or density function — it does not enclose an area of 1. Both functions are defined on different axes and therefore not directly comparable to each other PDF is defined on the data scale LF is defined on parameter scale

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Generalization © 2010 by John Fox York SPIDA n independent flips of a coin A particular sequence which includes x heads and n-x tails L( π |data)= Pr(data| π ) = ? We want the values of π that maximizes L( π |data) It is simpler, and equivalent to find the values of π that maximizes the log of the likelihood

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