Maximum Likelihood

Feb 01, 2014

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Topics. DefinitionsDemonstration in EXCELExample of ML in ASREML. Definition. Maximum = ?????????Likelihood = ?????????????ML = ????????? ?????????. x2 = 1225. X = ?????? 2 ??? 1225???????????? ?????? ?????? ??????? 122510?10 = 100 ?????????? ????????????100?100 = 10000 ????????? ????????????

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Maximum Likelihood

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Maximum likelihood estimation

Maximum likelihood estimation. Example: X 1 ,…,X n – i.i.d. random variables with probability p X (x|θ) = P(X=x) where θ is a parameter likelihood function L(θ|x) where x=(x 1 ,…,x n ) is set of observations maximum likelihood estimate maximizer of L(θ|x).

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Maximum Likelihood Estimation

Maximum Likelihood Estimation. Methods of Economic Investigation Lecture 17. Last Time. IV estimation Issues Heterogeneous Treatment Effects The assumptions LATE interpretation Weak Instruments Bias in Finite Samples F-statistics test. Today’s Class. Maximum Likelihood Estimators

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Maximum Likelihood Detection

Maximum Likelihood Detection. Dr. Muqaibel. Example. A binary repetition code is used where 0 is encoded as 000 and 1 is encoded as 111. What is your decoding decision if what you receive over a BSC with cross over probability=0.3 is 101. Using Minimum Distance Decoding.

784 views • 5 slides

Maximum likelihood (ML)

Maximum likelihood (ML). Conditional distribution and likelihood Maximum likelihood estimator Information in the data and likelihood Observed and Fisher’s information Home work. Introduction.

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Maximum Likelihood

Maximum Likelihood. There are three major paradigms of estimating linear models Method of Moments Oldest estimation method Population moments are best estimated by sample moments Not too useful for complex estimation Least Squares Minimize the sum of the squared errors

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Maximum Likelihood

Much estimation theory is presented in a rather ad hoc fashion. Minimising squared errors seems a good idea but why not minimise the absolute error or the cube of the absolute error?

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4. Maximum Likelihood

4. Maximum Likelihood. Prof. A.L. Yuille Stat 231. Fall 2004. Learning Probability Distributions. Learn the likelihood functions and priors from datasets. Two Main Strategies. Parametric and Non-Parametric. This Lecture and the next will concentrate on Parametric methods.

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Maximum Likelihood Estimation

Maximum Likelihood Estimation. Multivariate Normal distribution. The Method of Maximum Likelihood . Suppose that the data x 1 , … , x n has joint density function f ( x 1 , … , x n ; q 1 , … , q p )

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Maximum Likelihood Estimation

Maximum Likelihood Estimation. Psych 818 - DeShon. MLE vs. OLS. Ordinary Least Squares Estimation Typically yields a closed form solution that can be directly computed Closed form solutions often require very strong assumptions Maximum Likelihood Estimation

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Maximum Likelihood

Maximum Likelihood. We have studied the OLS estimator. It only applies under certain assumptions In particular, e ~ N(0, s 2 ) But what if the sampling distribution is not Normal? We can use an alternative estimator: MLE. See “Generalized Linear Models” in S-Plus. OLS vs. MLE.

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Maximum likelihood

Maximum likelihood. The maximum likelihood criterion. The optimal tree is that which would be most likely to give rise to the observed data (under a given model of evolution). An outline of the ML approach: Consider one character, i. (It is useful to arbitrarily root the tree).

559 views • 28 slides

Maximum likelihood decoding

Maximum likelihood decoding. v. r = v+ n. u. v’,u’. ECC encoder. ECC decoder. Channel. Decoder: Must determine v’ to minimize P(E| r )=P( v ’ v | r ) P(E) =  r P(E| r )P( r ) P( r ) is independent of decodingoptimum decoding must minimize P( v ’ v | r ) maximize P( v ’= v | r )

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Maximum likelihood (cont.)

Maximum likelihood (cont.). The maximum likelihood criterion. The optimal tree is that which would be most likely to give rise to the observed data (under a given model of evolution). Stepwise view. Propose a tree with branch lengths Consider the first character

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Maximum Likelihood Estimation

Learning. Probabilistic Graphical Models. Parameter Estimation. Maximum Likelihood Estimation. Biased Coin Example. P is a Bernoulli distribution: P(X=1) = , P(X=0) = 1- . Tosses are independent of each other Tosses are sampled from the same distribution (identically distributed).

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Maximum likelihood (cont.)

Maximum likelihood (cont.). Midterm. Relatedness – is not sharing a common ancestor but sharing a relatively recent CA Homoplasy includes convergence and/or reversal PTP looks at tree length ; g1 looks at skew in tree length Both introgression and LGT result in a dominant and minor history.

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Maximum Likelihood

Maximum Likelihood. Likelihood The likelihood is the probability of the data given the model. Likelihood. If we flip a coin and get a head and we think the coin is unbiased, then the probability of observing this head is 0.5.

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Maximum Likelihood

Maximum Likelihood. Benjamin Neale Boulder Workshop 2012. We will cover. Easy introduction to probability Rules of probability How to calculate likelihood for discrete outcomes Confidence intervals in likelihood Likelihood for continuous data. Starting simple.

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Maximum Likelihood

Flips usage of probability function A typical calculation: P(h|n,p) = C(h, n) * p h * (1-p) (n-h) The implied question: Given p of success in single trial, what is probability of h success over n trials?. Maximum Likelihood. The ML question. The ML calculation:

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Maximum Likelihood Estimation

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Maximum Likelihood Estimation

Maximum Likelihood Estimation. Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University. References: 1. Ethem Alpaydin, Introduction to Machine Learning , Chapter 4, MIT Press, 2004. Population (sample space). Sample. Inference. Statistics.

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Maximum Likelihood Estimate

Maximum Likelihood Estimate. Jyh-Shing Roger Jang ( 張智星 ) CSIE Dept, National Taiwan University. Intro. to Maximum Likelihood Estimate. MLE Maximum likelihood estimate Goal:

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Maximum Likelihood Detection

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