That means that the value of \(p\) that maximizes the natural logarithm of the likelihood function \(\ln L(p)\) is also the value of \(p\) that maximizes the likelihood function \(L(p)\). The log-likelihood can be written as follows:
(Note: the log-likelihood is closely related to information entropy and Fisher information. Remember, the parameter lambda is unknown and it is a parameter of the likelihood function. For instance,Again, the binomial distribution is the model to be worked with, with a single parameter ppp.
This vector is often called the score vector.
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Let us plot itThe above example shows that for our single data point the likelihood is the highest when lambda=2. For real-world problems, there are many reasons to avoid uniform priors. Its expected value is equal to the parameter μ of the given distribution,
which means that the maximum likelihood estimator
{\displaystyle {\widehat {\mu }}}
is unbiased. d data samples
y
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y
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{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})}
from some probability
y
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{\displaystyle y\sim P_{\theta _{0}}}
, that we try to estimate by finding
{\displaystyle {\hat {\theta }}}
that will maximize the likelihood using
P
{\displaystyle P_{\theta }}
, then:
Where
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{\displaystyle h_{\theta }(x)=\log {\frac {P(x\mid \theta _{0})}{P(x\mid \theta )}}}
. .