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L23-stats-post

L23-stats-post - X X Estimator W f W(w Readings Section 9.1...

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LECTURE 23 Readings: Section 9.1 (not responsible for t -based confidence intervals, in pp. 471-473) Outline Classical statistics Maximum likelihood (ML) estimation Estimating a sample mean Confidence intervals (CIs) CIs using an estimated variance Classical statistics Estimator ˆ Θ X N p X ( x ; θ ) θ also for vectors X and θ : p X 1 ,...,X n ( x 1 , . . . , x n ; θ 1 , . . . , θ m ) These are NOT conditional probabilities; θ is NOT random mathematically: many models, one for each possible value of θ Problem types: Hypothesis testing: H 0 : θ = 1 / 2 versus H 1 : θ = 3 / 4 Composite hypotheses: H 0 : θ = 1 / 2 versus H 1 : θ = 1 / 2 Estimation: design an estimator ˆ Θ , to keep estimation error ˆ Θ - θ small Maximum Likelihood Estimation Model, with unknown parameter(s): X p X ( x ; θ ) Pick θ that “makes data most likely” ˆ θ ML = arg max θ p X ( x ; θ ) Compare to Bayesian MAP estimation: ˆ θ MAP = max θ p X | Θ ( x | θ ) p Θ ( θ ) p X ( x ) Example: X 1 , . . . , X n : i.i.d., exponential( θ ) max θ n i =1 θ e - θ x
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