Likelihood function - PowerPoint PPT Presentation

Statistical Methods for Data Analysis parameter estimate

Statistical Methods for Data Analysis parameter estimate. Luca Lista INFN Napoli. Contents. Parameter estimates Likelihood function Maximum Likelihood method Problems with asymmetric errors. Meaning of parameter estimate. We are interested in some physical unknown parameters

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Pattern Recognition and Machine Learning

Pattern Recognition and Machine Learning. Chapter 1: Introduction. Example. Handwritten Digit Recognition. Polynomial Curve Fitting . Sum-of-Squares Error Function. 0 th Order Polynomial. 1 st Order Polynomial. 3 rd Order Polynomial. 9 th Order Polynomial. Over-fitting.

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Bayesian Estimators of Time to Most Recent Common Ancestry

Bayesian Estimators of Time to Most Recent Common Ancestry. Bruce Walsh. jbwalsh@u.arizona.edu. Ecology and Evolutionary Biology. Adjunct Appointments Molecular and Cellular Biology Plant Sciences Epidemiology &amp; Biostatistics Animal Sciences. Definitions.

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Christopher M. Bishop

Latent Variables, Mixture Models and EM. Christopher M. Bishop. Microsoft Research, Cambridge. BCS Summer School Exeter, 2003. Overview. K-means clustering Gaussian mixtures Maximum likelihood and EM Latent variables: EM revisited Bayesian Mixtures of Gaussians . Old Faithful.

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867 views • 55 slides

Addition of Independent Normal Random Variables

Addition of Independent Normal Random Variables . Theorem 1 : Let X and Y be two independent random variables with the N (  ,  2 ) distribution. Then, is N (2  , 2  2 ) . Proof : . Subtraction of Independent Normal Random Variables. Theorem 2 :

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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. 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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Pattern Recognition and Machine Learning

Source: Bishop book chapter 1 with modifications by Christoph F. Eick. Pattern Recognition and Machine Learning. Chapter 1: Introduction. Polynomial Curve Fitting . Experiment: Given a function; create N training example. What M should we choose?  Model Selection

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Stable Multi-Target Tracking in Real-Time Surveillance Video

Stable Multi-Target Tracking in Real-Time Surveillance Video. Active Vision Group Department of Engineering Science University of Oxford Ben Benfold Ian Reid. CVPR 2011. OUTLINE. Introduction Sliding Window Tracking Observations Data Association

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Effects of Viewing Geometry on Combination of Disparity and Texture Gradient Information

Effects of Viewing Geometry on Combination of Disparity and Texture Gradient Information. James M. Hillis. Michael S. Landy. Martin S. Banks. Outline. Background: Optimal cue combination Methods: slant discrimination Single-cue results Two-cue results: perceived slant

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Ordered probit models

Ordered probit models. Ordered Probit. Many discrete outcomes are to questions that have a natural ordering but no quantitative interpretation: Examples: Self reported health status (excellent, very good, good, fair, poor) Do you agree with the following statement

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Bayesian Adaptive Methods

Bayesian Adaptive Methods. Carrie Deis Nadine Dewdney. Overview. Phase I clinical trials Standard Designs Adaptive Designs Bayesian Approach Traditional vs. Bayesian Hybridization FDA Guidance Conclusion. Phase I . Conducted to determine toxicity for the dosing of the new intervention

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Models of migration Observations and judgments

Models of migration Observations and judgments. In: Raymer and Willekens, 2008, International migration in Europe, Wiley. Introduction:models. To interpret the world, we use models (mental schemes; mental structures) Models are representations of portions of the real world

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Factor Analysis

Factor Analysis. An Alternative technique for studying correlation and covariance structure. Let . have a p -variate Normal distribution . with mean vector . The Factor Analysis Model :. Let F 1 , F 2 , … , F k denote independent standard normal observations ( the Factors ).

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Econometrics 1

Econometrics 1. Lecture 7 Multicollinearity. What is multicollinearity. OLS Estimates in a Multiple Regression Model. Mutlicollinearity. Multicollinearity: A Numerical Example. Mutlicollinearity: In Matrix Notation. Mutlicollinearity: Cramer’s Rule for Estimation OLS Parameters.

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Lecture 17: Review of earlier concepts Regression for Case-control Studies

Lecture 17: Review of earlier concepts Regression for Case-control Studies. BMTRY 701 Biostatistical Methods II. The likelihood function. What is it? What is the log-likelihood function? Why are they important?. Maximum Likelihood:. What does it mean? What are MLEs? binomial model

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Consistency

Consistency. An estimator is a consistent estimator of θ, if , i.e., if converge in probability to θ. . Theorem. An unbiased estimator for θ, is a consistent estimator of θ if Proof:. Example.

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Empirical Estimator for GxE using imputed data

Empirical Estimator for GxE using imputed data. Shuo Jiao. Background.

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l Suppose we are trying to measure the true value of some quantity ( x T ). u We make repeated measurements of this

The M aximum L ikelihood M ethod (Taylor: “Principle of maximum likelihood”). l Suppose we are trying to measure the true value of some quantity ( x T ). u We make repeated measurements of this quantity { x 1 , x 2 , … x n }.

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Erin Anderson and David C. Schmittlein (1984)

Erin Anderson and David C. Schmittlein (1984) Integration of the sales force: an empirical examination Rand Journal of Economics. Research Question/Problem Formulation. OVERARCHING QUESTION: When does vertical integration occur? SPECIFIC QUESTION : Vertical integration of sales force

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269 views • 6 slides

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