Parametric statistical models

# Parametric statistical models - Parametric models I...

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Parametric statistical models 1 / 11 Parametric models A (statistical) model P = { P θ : θ Θ } is a family of probability distributions indexed by a set Θ (the parameter space ). In a parametric (statistical) model , the distributions are indexed by a finite number of parameters (i.e., Θ R k for some k < ). Examples: Bernoulli distributions on { 0 , 1 } Poisson distributions on Z + = { 0 , 1 , 2 , . . . } Gaussian distributions on R Multivariate Gaussian distributions on R d 2 / 11 Poisson distributions Poi( λ ) on Z + Indexed by positive scalar θ = λ > 0 called the rate parameter . The distribution P λ is supported (i.e., non-zero) on Z + . P λ ( x ) = exp( - λ ) λ x x ! . 3 / 11 Gaussian distributions N( μ, σ 2 ) on R Indexed by θ = ( μ, σ 2 ) : μ R is the mean , σ 2 > 0 is the variance . The distribution P μ,σ 2 has the probability density ϕ μ,σ 2 ( x ) = 1 2 πσ 2 exp - ( x - μ ) 2 2 σ 2 . 4 / 11

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Model fitting (parameter estimation) Suppose we have an observation x , and a parametric model P with parameter space Θ .
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