Statistical Inference Parametric Inference Maximum Likelihood Inference Exponential Families Expectation Maximization (EM) Bayesian Inference Statistical Decison Theory IP, José Bioucas Dias, IST, 2007 1 Statistical Inference Statistics aims at retriving the “causes” (e.g., parameters of a pdf) from the observations (effects) Probability Statistics Statistical inference problems can thus be seen as Inverse Problems As a result of this perpective, at the eighteenth century (at the time of Bayes and Laplace) Statistics was often called Inverse Probability IP, José Bioucas Dias, IST, 2007 2 Parametric Inference Consider the parametric model where is the parameter space and is the parameter The problem of inference reduces to the estimation of from ; i.e, Parameters of interest and nuisance parameters Let Sometimes we are only interested in some function that depends only on - parameter of interest; - nuisance parameter Example: IP, José Bioucas Dias, IST, 2007 3 Parametric Inference (theoretical limits) The Cramer Rao Lower Bound (CRLB) Under under appropriate regularity conditions, the covariance matrix of any Unbiased estimator , satisfies where is the Fisher information matrix given by An unbiased estimator that attains the CRLB may be found iif For some function h. The estimator is IP, José Bioucas Dias, IST, 2007 4 CRLB for the general Gaussian case Example: Parameter of a signal in white noise If Example: Known signal in unknown white noise IP, José Bioucas Dias, IST, 2007 5 Maximum Likelihood Method is the likelihood function If for all f we can use the log-likelihood Example (Bernoulli) IP, José Bioucas Dias, IST, 2007 6 Maximum Likelihood Example (Uniform) 1 1 IP, José Bioucas Dias, IST, 2007 7 Maximum Likelihood Example (Gaussian) IID Sample mean Sample variace IP, José Bioucas Dias, IST, 2007 8 Maximum Likelihood Example (Multivariate Gaussian) IID Sample mean Sample covariance IP, José Bioucas Dias, IST, 2007 9 Maximum Likelihood (linear observation model) Example: Linear observation in Gaussian noise A is full rank IP, José Bioucas Dias, IST, 2007 10 Example: Linear observation in Gaussian noise (cont.) • MLE is equivalent to the LSE using the norm • If , , is given by the Moore-Penrose Pseudo-Inverse • is a projection matrix (SVD) • If the noise is zero-mean but not Gaussian, the Best Linear Unbiased estimator (BLUE) is still given by IP, José Bioucas Dias, IST, 2007 11 Maximum likelihood Linear observation in Gaussian noise MLE Properties (MLE is optimal for the linear model) • Is the Minimum Variance Unbiased (MVU) estimator [ and is the minimum among all unbiased estimators] • Is efficient (it attains the Camer Rao Lower Bound (CRLB)) • Its PDF is IP, José Bioucas Dias, IST, 2007 12 Maximum likelihood (characterization) Appealing properties of MLE Let A sequence of IID vectors in and 1. The MLE is consistent: ( denotes the true parameter) 2. The MLE is equivariant: if is the MLE estimate of , then is the MLE estimate of 3. The MLE (under appropriate regularity conditions) is asymptotically Normal and optimal or efficient: Fisher information matrix IP, José Bioucas Dias, IST, 2007 13 The exponential Family Definition: the set an exponential family of dimension k if there there are functions such that is a sufficient statistic for f , i.e, Theorem: (Neyman-Fisher Factorization) is a sufficient statistic for f iif can be factored as IP, José Bioucas Dias, IST, 2007 14 The exponential family Natural (or canonical) form Given an exponential family, it is always possible to introduce the change of variables and the reparemeterization such that Since is a PDF, it must integrate to one IP, José Bioucas Dias, IST, 2007 15 The exponential family (The partition function) Computing moments from the derivatives of the partition function After some calculus IP, José Bioucas Dias, IST, 2007 16 The exponential family (IID sequences) Let a member of an exponential family defined by The density of the IID sequence is belongs exponential family defined by IP, José Bioucas Dias, IST, 2007 17 Examples of exponential families Many of the most common probabilistic models belong to exponential families; e.g., Gaussian, Poisson, Bernoulli, binomial, exponential, gamma, and beta. Example: Canonical form IP, José Bioucas Dias, IST, 2007 18 Examples of exponential families (Gaussian) Example: Canonical form IP, José Bioucas Dias, IST, 2007 19 Computing maximum likelihood estimates Very often the MLE can not be found analytically. Commonly used numerical methods: 1. Newton-Raphson 2. Scoring 3. Expectation Maximization (EM) Newton-Raphson method Scoring method Can be computed off-line IP, José Bioucas Dias, IST, 2007 20 Computing maximum likelihood estimates (EM) Expectation Maximization (EM) [Dempster, Laird, and Rubin, 1977] Suppose that is hard to maximize But we can find a vector z such that is easy to maximze and Idea: iterate between two steps: E-step: “Fill in z” in Terminology Observed data M-step: Maximize Missing data Complete data IP, José Bioucas Dias, IST, 2007 21 Expectation maximization The EM algorithm 1. Pick up a starting vector : repeat steps 2. and 3. 2. E-step: Calculate 3. M-step Alternatively (GEM) IP, José Bioucas Dias, IST, 2007 22 Expectation maximization The EM (GEM) algorithm always increases the likelihood. Define 1. 2. Kulback Leibler distance 3. 4. KL distance maximization IP, José Bioucas Dias, IST, 2007 23 Expectation maximization (why does it work?) IP, José Bioucas Dias, IST, 2007 24 EM: Mixtures of densities Let be the random variable that selects the active mode: where and IP, José Bioucas Dias, IST, 2007 25 EM: Mixtures of densities Consider now that is a sequence of IID random variables Let be IID random variables, where selects the active mode in the sample : IP, José Bioucas Dias, IST, 2007 26 EM: Mixtures of densities Equivalent Q Where is the sample mean of x, i.e., IP, José Bioucas Dias, IST, 2007 27 EM: Mixtures of densities E-step: M-step: IP, José Bioucas Dias, IST, 2007 28 EM: Mixtures of densities E-step: M-step: IP, José Bioucas Dias, IST, 2007 29 EM: Mixtures of Gaussian densities (MOGs) E-step: M-step: Weighted sample mean Weighted sample covariance IP, José Bioucas Dias, IST, 2007 30 EM: Mixtures of Gaussian densities. 1D Example 0 1 0.6316 -0.0288 1.0287 0.6258 3 3 0.3158 2.8952 2.5649 0.3107 6 10 0.0526 6.1687 7.3980 0.0635 p = 3 loglikelihood L(fk) N = 1900 -3800 -4000 -4200 -4400 -4600 -4800 -5000 -5200 0 5 10 15 20 25 30 IP, José Bioucas Dias, IST, 2007 31 EM: Mixtures of Gaussian Densities (MOGs) 0 1 0.6316 p = 3 Example – 1D 3 3 0.3158 N = 1900 6 10 0.0526 0.35 0.35 hist hist est MOG est modes 0.3 0.3 true MOG true modes 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -5 0 5 10 15 -6 -4 -2 0 2 4 6 8 10 12 IP, José Bioucas Dias, IST, 2007 32 EM: Mixtures of Gaussian Densities: 2D Example MOG with determination of the number of modes [M. Figueiredo, 2002] k=3 2 0 -2 -2 0 2 4 IP, José Bioucas Dias, IST, 2007 33 Bayesian Estimation IP, José Bioucas Dias, IST, 2007 34 The Bayesian Philosophy ([Wasserman, 2004]) Bayesian Inference B1 – Probabilities describe degrees of belief, not limiting relative frequency B2 – We can make probability statements about parameters, even though they are fixed parameters B3 – We make inferences about a parameter by producing a probalility distribution for Frequencist or Classical Inference F1 – Probabilities refer to limiting relative frequencies and are objective properties of the real world F2 – Parameters are fixed unknown parameters F3 – The criteria for obtaining statistical procedures are based on long run frequency properties. IP, José Bioucas Dias, IST, 2007 35 The Bayesian Philosophy Observation model unknown observation Prior knowledge Bayesian Inference Classical Inference describes degrees of belief (subjective), not limiting frequency IP, José Bioucas Dias, IST, 2007 36 The Bayesian method 1. Choose a prior density , called the prior (or a priori) distribution that expresses our beliefs about f, before we see any data 2. Choose the observation model that reflects our belief about g given f 3. Calculate the posterior (or a posteriori) distribution using the Bayes law: where is the marginal on g (other names: evidence, unconditional, predictive) 4. Any inference should be based on the posterior IP, José Bioucas Dias, IST, 2007 37 The Bayesian method Example: Let IID and 4 ==0.5 for = >1, pulls 3.5 ==1 ==2 towards 1/2 3 ==10 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IP, José Bioucas Dias, IST, 2007 38 Example (cont.): (Bernoulli observations, Beta prior) Observation model Prior Posterior Thus, IP, José Bioucas Dias, IST, 2007 39 Example (cont.): (Bernoulli observations, Beta prior) Maximum a posteriori estimate (MAP) • Total ignorance: flat prior = =1 • Note that for large values of n The von Mises Theorem If the prior is continuous and not zero at the location of the MLestimate, then, IP, José Bioucas Dias, IST, 2007 40 Conjugate priors In the previous example, the prior and the posterior are both Beta distributed. We say that the prior is conjugate with respect to the model • Formally, let and be two parametrized families of priors and observation models, respectively • is a conjugate family for if for some • Very often, prior information about f is very small, allowing to select conjugate priors • Conjugate priors why? Computing the posterior density simply consists in updating the parameters of the prior IP, José Bioucas Dias, IST, 2007 41 Conjugate priors (Gaussian observation, Gaussian prior) • Gaussian observations • Gaussian prior • The posterior distribution is Gaussian 1.