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% BB Aug 28, 2005.
% BB Dec 16, 2004.

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% \date{\color{blue} Leipzig University, December 16, 2004}
\date{\color{blue} FSU, August 29, 2006}
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% ------------------------------------------------------------------
% {\r From Popular Mechanics, 1954:}
% \begin{figure} \begin{center}
% \includegraphics[width=0.7\textwidth]{figures/MCMC01.jpg}
% \caption{\small Book cover.}
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\title{\r Markov Chain Monte Carlo Simulations and Their Statistical 
          Analysis -- An Overview }
\author{\color{blue} Bernd Berg}
\maketitle
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\foilhead{\Large\bf\r Content} \b \vskip -20pt

\begin{enumerate}
\item Statistics as needed
\item Markov Chain Monte Carlo (MC)
\item Statistical Analysis of MC Data and Advanced MC.
\end{enumerate}

%\noindent\b Link to {\r computer code} at www.scs.fsu.edu/\~\ $\!$berg.

\vfil

\foilhead{\r Probability Distributions and Sampling}\b\vskip -20pt

\noindent In $N$ experiments we may find an event $A$ to occur $n$ 
times. The {\bf frequency definition} of the {\bf probability} of
the event is \s
$$ P(A)\ =\ \lim_{N\to\infty} {n\over N}\ . $$ \b
Let $P(a,b)$ be the probability that $x^r\in [a,b]$ where $x^r$ is a 
{\bf random variable} drawn in the interval $(-\infty,+\infty)$ with 
a {\bf probability density} $f(x)>0$. Then, \s
$$ P(a,b) = \int_a^b dx\, f(x) ~~~{\rm and}~~~
   f(x) = \lim_{y\to x} {P(y,x) \over x-y}\ . $$ \b
The {\bf (cumulative) distribution function} of the random variable 
$x^r$ is defined as \s
$$ F(x) = P(x^r\le x) = \int_{-\infty}^x f(x')\,dx'\ .$$ \b
For {\bf uniform probability distribution} between \s $[0,1)$, 
$$ u(x) = \cases{ 1~~{\rm for}~~0\le x < 1;\cr
                  0 ~~{\rm elsewhere}. } $$ \b
The corresponding distribution function is \s
$$ U(x) = \int_{-\infty}^x u(x')\,dx' = \cases{
   0~~{\rm for}~~x<0;\cr
   x~~{\rm for}~~0\le x\le 1;\cr 1~~{\rm for}~~x>1.} $$ \b
It allows for the construction of general probability distributions.  
Let \s
$$ y = F(x) = \int_{-\infty}^x f(x')\,dx'\ . $$ \b
For \s $y^r$ \b being a uniformly distributed random variable in \s $[0,1)$ 
$$ x^r = F^{-1} (y^r)~~{\rm\b is\ then\ distributed\ according\ to\
   the\ probability\ density}~~f(x)\ . $$

\foilhead{\r Pseudo Random Numbers and Computer Code} \b \vskip -20.0pt

\noindent It is sufficient to generate uniform (pseudo) random numbers.
{\r Control your random number generator!} Therefore, a portable,
well-tested generator should be choosen. My code supplies a generator
by Marsaglia and collaborators whith an approximate period of \s
$2^{110}$. \b How to get it? Download {\tt\s STMC.tgz} which unfolds
under (Linux) \hfil\break
\centerline{\tt\s tar -zxvf STMC.tgz}
into the directory structure shown below.

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\foilhead{\r Statistical Topics to Follow:}\b \vskip -20pt

{\large Confidence Intervals and Sorting}

{\large Central Limit Theorem:}  
Convergence to Gaussian Sample Mean.

{\large Binning (Blocking) and Jackknife Error Analysis} 

{\large Various Difference Tests} 

{\large Error of the Error Bar}

{\large Determination of Parameters (Fitting)}


\foilhead{\r Statistical Physics and Markov Chain Monte Carlo 
             Simulations} \vskip -30pt

\noindent \b MC simulations of systems described by the Gibbs canonical 
ensemble aim at calculating estimators of physical observables at a 
temperature $\s T$. In the following we consider the calculation of 
the {\r expectation value} of an {\r observable} $\s \mathcal{O}$. All 
systems on a computer are discrete, because a finite word length has 
to be used. Hence, 
{\s \begin{eqnarray} \nonumber
  \widehat{\mathcal{O}} = \widehat{\mathcal{O}} (\beta) = 
  \langle \mathcal{O} \rangle &=& Z^{-1} \sum_{k=1}^K
  \mathcal{O}^{(k)}\,e^{-\beta\,E^{(k)} }\\ \nonumber
  {\rm where}~~~
  Z\ =\ Z(\beta) &=& \sum_{k=1}^K e^{-\beta\,E^{(k)} }
\end{eqnarray} }
is the {\r partition function}. The index $\s k=1,\dots , K$ labels all 
{\r configurations (or microstates)} of the system, and $\s E^{(k)}$ is 
the (internal) {energy } of configuration $\s k$. 
\medskip

\centerline{No direct way to generate the {\r important}
configuration!}
% \noindent {\r Continuous systems} are also treated in the book.


\foilhead{\r Markov Chain Monte Carlo}
\vskip -30pt

\noindent A Markov chain allows to generate configurations $\s k$
with probability
$$\s  P_B^{(k)} = c_B\, w^{(k)}_B = c_B\, e^{-\beta E^{(k)}}\,,
   \qquad c_B~~{\rm constant}\,. $$
The {\r state vector} $\s (P_B^{(k)})$, for which the configurations are 
the vector indices, is called {\r Boltzmann state}. A Markov chain is a 
simple dynamic process, which generates configuration $\s k_{n+1}$
stochastically from configuration $\s k_n$. Let the {\r transition
probability} to create the configuration $\s l$ in one step from $\s k$ 
be given by $\s W^{(l)(k)} = W[k\to l]$. Then, the transition matrix
$$\s W = \left( W^{(l)(k)}\right) $$
defines the Markov process. Note, that this matrix is a very big 
and never stored in the computer. The matrix achieves our goal and 
generates configurations with the desired probabilities, when it
satisfies certain properties.

\newpage

\centerline{\r\large Many Algorithms!}

Metropolis, Heatbath, cluster, ...  \bigskip \bigskip

\centerline{\r\large Statistical Errors of Markov Chain MC Data}

\b \noindent A typical MC simulation falls into two parts: 

\begin{enumerate}

\item Equilibration without measurements.

\item Production with measurements.

\end{enumerate}

\centerline{\r\large Take Autocorrelations into Account!}

\centerline{$\Rightarrow$ Comparison of Markov chain MC algorithms.}
\bigskip \bigskip

\centerline{\r\large Generalized Ensembles}

Replica exchange method (parallel computing, MPI), multicanonical, ...

\foilhead{\r\Large\bf Summary}

\begin{itemize}
\item We intend to consider Statistics, Markov Chain Monte Carlo 
      simulations, the Statistical Analysis of Markov chain data and, 
      finally, generalized ensembles and parallel computing (MPI).

\item Each method comes with its own Fortan code (if you insist
      on C or C++ say it now). 

\end{itemize}

\begin{center} \large\r 
We will primarily train to get things up and running\\ 
for interesting applications.
\end{center}
\vfil

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