mw-parser-output .
This inequality, called information inequality by many
authors, is essential for proving the consistency of the maximum likelihood
estimator. Specifically,18
where
I
{\displaystyle ~{\mathcal {I}}~}
is the Fisher information matrix:
In particular, it means that the bias of the maximum likelihood estimator is equal to zero up to the order . And, the last equality just uses the shorthand mathematical notation of a product of indexed terms.
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By applying Bayes’ theorem
and if we further assume the zero-or-one loss function, which is a same loss for all errors, the Bayes Decision rule can Our site reformulated as:
where
Bayes
{\displaystyle h_{\text{Bayes}}}
is the prediction and
(
w
)
{\displaystyle \;\operatorname {\mathbb {P} } (w)\;}
is the prior probability. getFullYear()) Aptech Systems, Inc.
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write(new Date(). 7 For an open
{\displaystyle \,\Theta \,}
the likelihood function may increase without ever reaching a supremum value. . In doing so, we’ll use a “trick” that often makes the differentiation a bit easier.
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Suppose the coin is tossed 80 times: i. This result is
equivalent to the result we need to prove
because
The latter next is often called information equality. But how would we implement the method in practice? Well, suppose we have a random sample \(X_1, X_2, \cdots, X_n\) for which the probability density (or mass) function of each \(X_i\) is \(f(x_i;\theta)\). Further, if the function
n
:
R
n
{\displaystyle \;{\hat {\theta }}_{n}:\mathbb {R} ^{n}\to \Theta \;}
so defined is measurable, then it is called the maximum likelihood estimator.
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.