Maximum-Likelihood and Bayesian Parameter Estimationz Sufficient Statisticsz Common Probability Distributionsz Problems of DimensionalitySufficient Statisticsz Any function s of the samples is a statisticz A sufficient statistic is a (possibly vector valued) function s of samples D that contains all information relevant to estimating some parameter θFactorization Theorem:A statistic s is sufficient for θiff the probability P(D| θ)can be written as the productP(D| θ) = g(s, θ) h(D)For some functions g(. , .) and h(. , .)Sufficient Statistics and Gaussian),(~)|(ΣθθNxpUnknown meanIsolates dependence on θin the first term⎥⎦⎤⎢⎣⎡Σ−Σ⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛Σ+Σ=⎥⎦⎤⎢⎣⎡−Σ−−Σ=∑∑∏=−=−−−=nkktknndnkkttktknkdxxxxxDp112/2/111112/12/21exp ||)2(1 2n-exp )()(21exp||)2(1)|(πθθθθθπθxSufficient StatisticCommon Exponential Distributionsand their Sufficient StatisticsName Distribution Domain s[g(s,θ)]1/nCommon Exponential Distributionsand their Sufficient StatisticsDistributionName Domain
View Full Document
Unlocking...