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Chapter 15. Chapter 15 Maximum Likelihood Estimation, Likelihood Ratio Test, Bayes Estimation, and Decision Theory. Bei Ye, Yajing Zhao, Lin Qian, Lin Sun, Ralph Hurtado, Gao Chen, Yuanchi Xue, Tim Knapik, Yunan Min, Rui Li. Section 15.1. Maximum Likelihood Estimation.

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## Chapter 15

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**Chapter 15**Chapter 15 Maximum Likelihood Estimation, Likelihood Ratio Test, Bayes Estimation, and Decision Theory Bei Ye, Yajing Zhao, Lin Qian, Lin Sun, Ralph Hurtado, Gao Chen, Yuanchi Xue, Tim Knapik, Yunan Min, Rui Li**Section 15.1**Maximum Likelihood Estimation**Maximum Likelihood Estimation (MLE)**• Likelihood function • Calculation of MLE • Properties of MLE • Large sample inference and delta method • 2.Calculation of Maximum Likelihood Estimation • 2.Calculation of Maximum Like**Likelihood Function**1.1 Parameter space Θ X1 ,…, Xn : i.i.d. observations θ: an unknown parameter Θ: The set of all possible values of θ 1.2 Joint p.d.f. or p.m.f. of X1 ,…, Xn**Likelihood Function**1.3 Likelihood Function of θ For observed χ1,…,χn: • The joint p.d.f. or p.m.f. is a function of χ1,…,χnfor given θ . • The likelihood function is a function of θ for given χ1,…,χn.**Example : Normal Distribution**Suppose χ1,…,χn is a random sample from a normal distribution with p.d.f.: a vector parameter: ( ) • Likelihood Function:**Calculation of MLE**2.1 Maximum Likelihood Estimation: Need to find which maximizes the likelihood function . • Simple Example: • Two independent Bernoulli trials with success probabilityθ. • θ is known : 1/4 or 1/3 (Θ). • The probabilities of observing χ= 0, 1, 2 successes can be calculated. Let’s look at the following table.**Probability of ObservingχSuccesses**The MLE is chosen to maximize for observed χ. χ=0:χ=1 or 2: Calculation of MLE**Calculation of MLE**2.2 Log-likelihood function: Setting the derivative of L(θ) equal to zero and solving for θ. • Note: The likelihood function must be differentiable and then this method can be used.**Properties of MLE**MLE: optimality properties in large samples The concept of information due to Fisher**Properties of MLE**3.1 Fisher Information: • Alternative expression**Properties of MLE**• for an i.i.d. sample:**Properties of MLE**• for k-dimensional vector parameter**Properties of MLE**3.2 Cramér-Rao Lower Bound • A random sample X1, X2, …, Xn from p.d.f f(x|θ). • Let be any estimator of θ with where B(θ) is the bias of If B(θ) is differentiable in θ and if certain regularity conditions holds, then • (Cramér-Rao inequality) • The ratio of the lower bound to the variance of any estimator of θ is called the efficiency of the estimator. • An estimator has efficiency = 1 is called the efficient estimator.**4.1 Large Sample Inferences**To make Large sample inference on unknown parameter θ (Single Parameter), we need to estimate : I(θ) is estimated by: This estimate does not require evaluation of the expected value. An approximate large sample CI on θ is: Large Sample Inferences and Delta Method**4. Large Sample Inferences and Delta Method**4.2 Delta Method for Approximating the Variance of an Estimator To estimate a nonlinear function h(θ). Suppose that : and is a known function of θ Delta Method:**Section 15.2**Likelihood Ratio Test**Likelihood Ratio (LR) Test**• Background of LR test • Neyman-Pearson Lemma and Test • Examples • Generalized Likelihood Ratio Test**Background of LR test**Egon Sharpe Pearson1895-1980English mathematician • Jerzy Splawa Neyman, 1894-1981 Polish-American mathematician.**Neyman-Pearson lemma**We want to find a rejection region R such that the error of both type I and type II error are as small as possible. Suppose have joint p.d.f Consider the ratio Then a best critical region of is Where is a constant such that**What is Likelihood Ratio (LR) test**• A ratio is computed between the maximum probability of a result under null and alternative hypothesis. where the numerator corresponds to the maximum probability of an observed result under the null hypothesis; denominator under the alternative hypothesis. • Test idea: if observe x, then condition is evidence in favor of the alternative; the opposite inequality is evidence against the alternative. • Hence, the decision of rejecting null hypothesis was made based on the value of this ratio.**The Test**• Let be a random sample with p.d.f • Hypothesis: • Test statistic: • Reject when the likelihood ratio exceeds**Characteristics of LR Test**• Most powerful test of significance level α; Maximize • Very useful and widely applicable, esp. in medicine to assist in interpreting diagnostic tests . • Exact distribution of the likelihood ratio corresponding to specific hypotheses is very difficult to determine. • The computations are difficult to perform by hand.**Example 1: Test on Normal Distribution Mean**The likelihoods under and are:**Example 1 continued**Likelihood ratio is Reject when the ratio exceeds a constant , which is chosen for a specific significance level: The test is independent of . It is the most powerful level test for all .**A Numerical Example**Suppose a random sample with size and Test versus with Where So we reject if .**Generalized Likelihood Ratio Test**• Neyman-Pearson Lemma shows that the most powerful test for a Simple vs. Simple hypothesis testing problem is a Likelihood Ratio Test. • We can generalize the likelihood ratio method for the Composite vs. Composite hypothesis testing problem.**Hypothesis**• Suppose H0 specifies that θ is in Θ0 and H1 specifies that θ is in Θ0c. Symbolically, the hypotheses are:**Test Statistics**• Note that λ ≤ 1. • An intuitive way to understand λ is to view: • the numerator of λ as the maximum probability of the observed sample computed over parameters in the null hypothesis • the denominator of λ as the maximum probability of the observed sample over allpossible parameters. • λ is the ratio of these two maxima**Test Statistics**• If H0 is true, λ should be close to 1. • If H1 is true, λ should be smaller. • A small λ means that the observed sample is much more likely for the parameter points in the alternative hypothesis than for any parameter point in the null hypothesis.**Reject Region & Critical Constant**• Rejects Ho if λ < k, where k is the critical constant. • k < 1. • k is chosen to make the level of the test equal to the specified α, that is, α = PΘo ( λ ≤ k ).**A Simple Example to illustrate GLR test**• Ex 15.18 (GLR Test for Normal Mean: Known Variance) • For a random sample x1, x2……, xnfrom an N (μ, σ2) distribution with known σ2, derive the GLR test for the one-sided testing problem: Ho: µ ≤ µ0 vs. H1: µ > µ0 where µ0 is specified.**Solutions**• If , then the restricted MLE of µ under H0 is simply . • If , then the restricted MLE of µ under H0 is , because in this case, the maximum of the likelihood function under H0 is attained at .**SolutionsThus, the numerator & denominator of the likelihood**ratio are showing below, respectively**SolutionTaking the ratio of the two and canceling the common**terms, we get**Solution**• Clearly, we do not reject H0 when λ = 1, i.e., when . • Therefore, the condition λ < k is equivalent to subject to . • In other words, we reject H0 if is large, which leads to the usual upper one sided z-test.**Section 15.3**Bayesian Inference**Bayesian Inference**• Background of Bayes • Bayesian Inference defined • Bayesian Estimation • Bayesian Testing**Background of Thomas Bayes**• Thomas Bayes • 1702 – 1761 • British mathematician and Presbyterian minister • Fellow of the Royal Society • Studied logic and theology at the University of Edinburgh • He was barred from studying at Oxford and Cambridge because of his religion**Background of Bayes**• Baye’s Theorem • Famous probability theorem for finding “reverse probability” • The theorem was published posthumously in a paper entitled “Essay Towards Solving a Problem in the Doctrine of Chances”**Bayesian Inference**• Application to Statistics – Qualitative Overview • Estimate an unknown parameter • Assumes the investigator has some prior knowledge of the unknown parameter • Assumes the prior knowledge can be summarized in the form of a probability distribution on , called the prior distribution • Thus, is a random variable**Bayesia**• Application to Statistics – Qualitative Overview (cont.) • The data are used to update the prior distribution and obtain the posterior distribution • Inferences on are based on the posterior distribution**Bayesian Inference**• Criticisms by Frequentists • Prior knowledge is not accurate enough to form a meaningful prior distribution • Perceptions of prior knowledge differ from person to person • This may cause inferences on the same data to differ from person to person.**Some Key Terms in Bayesian Inference…**In the classical approach the parameter, θ, is thought to be an unknown, but fixed, quantity. In the Bayesian approach, θ is considered to be a quantity whose variation can be described by a probability distribution which is called prior distribution. • prior distribution – a subjective distribution, based on experimenter’s belief, and is formulated before the data are seen. • posterior distribution – is computed from the prior and the likelihood function using Bayes’ theorem. • posterior mean – the mean of the posterior distribution • posterior variance – the variance of the posterior distribution • conjugate priors - a family of prior probability distributions in which the key property is that the posterior probability distribution also belongs to the family of the prior probability distribution**1.5.3.1 Bayesian Estimation**Now lets move on to how we can estimate parameters using Bayesian approach. Now let’s move on to how we can estimate parameters using Bayesian approach. (Using text notation) Let be an unknown parameter based on a random sample, from a distribution with pdf /pmf Let be the prior distribution of Let be the posterior distribution If we apply Bayes Theorem(Eq. 15.1), our posterior distribution becomes : Note that is the marginal PDF of X1,X2,…Xn**Bayesian Estimation(continued)**As seen in equation 15.2, the posterior distribution represents what is known about after observing the data . From earlier chapters, we know that the likelihood of a variable is So, to get a better idea of the posterior distribution, we note that: posterior distribution likelihood prior distribution i.e. For a detailed practical example of deriving the posterior mean and using Bayesian estimation, visit: http://www.stat.berkeley.edu/users/rice/Stat135/Bayes.pdf**Example 15.25**Let x be an observation from an distribution where μ is unknown and σ2 is known. Show that the normal distribution is a conjugate prior on μ. We can ignore the factor because it will cancel from both the numerator and denominator of the expression for . Similarly, any terms not involving μ can be canceled from the numerator and denominator.**Example 15.25 (continue)**Thus, we see that is proportional to Where and It follows that has the form of the normal distribution. Specifically, is distribution (the normalizing constant comes from the denomination.

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