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Pattern Recognition by Matthias Nagel, Matthias Richter, Jürgen Beyerer

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is called the maximum-likelihood estimator (ML estimator).

Under the usual implicit assumption that all functions are sufficiently smooth,

0 1 = θ l( θ )= mD θ ln ( p( m|θ ) )with θ = ( θ 1 θ q ) ( 4.29 )

is a necessary condition.

The first example will be to find the ML estimator for the expectation value of a d-dimensional normal distribution. Let mk ∼ N(μ, Σ) with μ unknown but known Σ. It follows that

lnp( m k )= 1 2 ( m k μ ) 1 ( m k μ ) d 2 ln2π 1 2 lndetΣ μ lnp( m k )= 1 ( m k μ ) .( 4.30 )

Applying ...

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