teaching/scientificComputing
Archived
3
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

fixed many index entries

Browse Source
This commit is contained in:
Jan Benda
2019-12-09 20:01:27 +01:00
parent f24c14e6f5
commit bf52536b7b
12 changed files with 332 additions and 306 deletions
Download Patch File Download Diff File Expand all files Collapse all files

24
likelihood/lecture/likelihood.tex
View File

@@ -26,15 +26,16 @@ parameters $\theta$. This could be the normal distribution
defined by the mean $\mu$ and the standard deviation $\sigma$ as
parameters $\theta$. If the $n$ independent observations of $x_1,
x_2, \ldots x_n$ originate from the same probability density
distribution (they are \enterm{i.i.d.} independent and identically
distributed) then the conditional probability $p(x_1,x_2, \ldots
distribution (they are \enterm[i.i.d.|see{independent and identically
distributed}]{i.i.d.}, \enterm{independent and identically
distributed}) then the conditional probability $p(x_1,x_2, \ldots
x_n|\theta)$ of observing $x_1, x_2, \ldots x_n$ given a specific
$\theta$ is given by
\begin{equation}
p(x_1,x_2, \ldots x_n|\theta) = p(x_1|\theta) \cdot p(x_2|\theta)
\ldots p(x_n|\theta) = \prod_{i=1}^n p(x_i|\theta) \; .
\end{equation}
Vice versa, the \enterm{likelihood} of the parameters $\theta$
Vice versa, the \entermde{Likelihood}{likelihood} of the parameters $\theta$
given the observed data $x_1, x_2, \ldots x_n$ is
\begin{equation}
{\cal L}(\theta|x_1,x_2, \ldots x_n) = p(x_1,x_2, \ldots x_n|\theta) \; .
@@ -57,7 +58,7 @@ The position of a function's maximum does not change when the values
of the function are transformed by a strictly monotonously rising
function such as the logarithm. For numerical and reasons that we will
discuss below, we commonly search for the maximum of the logarithm of
the likelihood (\enterm{log-likelihood}):
the likelihood (\entermde[likelihood!log-]{Likelihood!Log-}{log-likelihood}):
\begin{eqnarray}
\theta_{mle} & = & \text{argmax}_{\theta}\; {\cal L}(\theta|x_1,x_2, \ldots x_n) \nonumber \\
@@ -136,9 +137,10 @@ from the data.
For non-Gaussian distributions (e.g. a Gamma-distribution), however,
such simple analytical expressions for the parameters of the
distribution do not exist, e.g. the shape parameter of a
\enterm{Gamma-distribution}. How do we fit such a distribution to
some data? That is, how should we compute the values of the parameters
of the distribution, given the data?
\entermde[distribution!Gamma-]{Verteilung!Gamma-}{Gamma-distribution}. How
do we fit such a distribution to some data? That is, how should we
compute the values of the parameters of the distribution, given the
data?
A first guess could be to fit the probability density function by
minimization of the squared difference to a histogram of the measured
@@ -289,10 +291,10 @@ out of \eqnref{mleslope} and we get
To see what this expression is, we need to standardize the data. We
make the data mean free and normalize them to their standard
deviation, i.e. $x \mapsto (x - \bar x)/\sigma_x$. The resulting
numbers are also called \enterm[z-values]{$z$-values} or $z$-scores and they
have the property $\bar x = 0$ and $\sigma_x = 1$. $z$-scores are
often used in Biology to make quantities that differ in their units
comparable. For standardized data the variance
numbers are also called \entermde[z-values]{z-Wert}{$z$-values} or
$z$-scores and they have the property $\bar x = 0$ and $\sigma_x =
1$. $z$-scores are often used in Biology to make quantities that
differ in their units comparable. For standardized data the variance
\[ \sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 = 1 \]
is given by the mean squared data and equals one.
The covariance between $x$ and $y$ also simplifies to