Monte Carlo Methods in Statistical Mechanics: Foundations and ...

is very far from equilibrium. By throwing away the data from the initial transient, we
lose nothing, and avoid a potentially large systematic error.
Autocorrelation in equilibrium. As explained in the preceding lecture, the variance
of the sample mean f inadynamic Monte Carlomethod is a factor 2 intf higher than
it would be in independent sampling. Otherwise put, a run oflength n contains only
n=2 intf \e ectively independent data points".
This has several implications for Monte Carlo work. On the onehand, it means
that the the computational e ciency of the algorithm is determined principally by its
autocorrelation time. More precisely, ifonewishes to compare two alternative Monte
Carlo algorithms for the same problem, then the better algorithm is the one that has
the smaller autocorrelation time, when time is measured in units of computer (CPU)
time. [In general there may arise tradeo s between \physical" autocorrelation time (i.e.
measured in iterations) and computational complexity per iteration.] So accurate
measurements ofthe autocorrelation time are essential to evaluating the computational
e ciency of competing algorithms.
On the other hand, even for a xed algorithm, knowledge of intf is essential for
determining run lengths | is a run of 100000 sweeps long enough? | and for setting
error bars on estimates of hfi. Roughly speaking, error bars will be of order ( =n) 1=2
so if we want 1% accuracy, then we need a run oflength 10000 ,andsoon. Above
all, there is a basic self-consistency requirement: the runlength n must be than the
estimates of produced by that samerun, otherwise none of the results fromthat run
should be believed. Of course, while self-consistency is a necessary condition for the
trustworthiness of Monte Carlo data, it is not a su cient condition there is always the
danger of metastability.
Already we can draw a conclusion about the relative importance of initialization
bias and autocorrelation as di culties in dynamic Monte Carlo work. Let us assume
that the time for initial convergence to equilibrium is comparable to (orat least not too
much larger than) the equilibrium autocorrelation time intf (for the observables f of
interest) | thisisoften but notalways the case. Then initialization bias is a relatively
trivial problem compared to autocorrelation in equilibrium. To eliminate initialization
bias, it su ces to discard 20 of the data atthe beginning ofthe run but toachieve
a reasonably small statistical error, it is necessary to make arunoflength 1000 or
more. So the data that must be discarded at the beginning, ndisc, isanegligible fraction
of the total run length n. This estimate alsoshows that the exact value of ndisc is not
particularly delicate: anything between 20 and n=5 will eliminate essentially all
initialization bias while paying less than a 10% price in the nal error bars.
In this remainder of this lecture I would like to discuss in more detail the statistical
analysis of dynamic Monte Carlo data (assumed to be already \in equilibrium"), with
emphasis on how toestimate the autocorrelation time intf and how tocompute valid
error bars. What isinvolved here is a branch ofmathematical statistics called timeseries
analysis. An excellent exposition can be found inthe books of of Priestley [14]
and Anderson [15].
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