Heidelberger and Welch (1981; 1983) proposed a two-part MCMC convergence diagnostic that calculates a test statistic (based on the Cramer-von Mises test statistic) to accept or reject the null hypothesis that the Markov chain is from a stationary distribution.

`Heidelberger.Diagnostic(x, eps=0.1, pvalue=0.05)`

x

This required argument accepts an object of class
`demonoid`

. It attempts to use `Posterior2`

, but when this
is missing it uses `Posterior1`

.

eps

This argument specifies the target value for the ratio of halfwidth to sample mean.

pvalue

This argument specifies the level of statistical significance.

The `Heidelberger.Diagnostic`

function returns an object of class
`heidelberger`

. This object is a \(J \times 6\)
matrix, and it is intended to be summarized with the
`print.heidelberger`

function. Nonetheless, this object of
class `heidelberger`

has \(J\) rows, each of which corresponds
to a Markov chain. The column names are `stest`

, `start`

,
`pvalue`

, `htest`

, `mean`

, and `halfwidth`

. The
`stest`

column indicates convergence with a one, and
non-convergence with a zero, regarding the stationarity test. When
non-convergence is indicated, the remaining columns have missing
values. The `start`

column indicates the starting iteration, and
the `pvalue`

column shows the p-value associated with the first
test. The `htest`

column indicates convergence for the halfwidth
test. The `mean`

and `halfwidth`

columns report the mean and
halfwidth.

The Heidelberg and Welch MCMC convergence diagnostic consists of two parts:

First Part 1. Generate a chain of \(N\) iterations and define an alpha level. 2. Calculate the test statistic on the whole chain. Accept or reject the null hypothesis that the chain is from a stationary distribution. 3. If the null hypothesis is rejected, then discard the first 10% of the chain. Calculate the test statistic and accept or reject the null hypothesis. 4. If the null hypothesis is rejected, then discard the next 10% and calculate the test statistic. 5. Repeat until the null hypothesis is accepted or 50% of the chain is discarded. If the test still rejects the null hypothesis, then the chain fails the test and needs to be run longer.

Second Part If the chain passes the first part of the diagnostic, then the part of the chain that was not discarded from the first part is used to test the second part.

The halfwidth test calculates half the width of the (1 - alpha)% probability interval (credible interval) around the mean.

If the ratio of the halfwidth and the mean is lower than `eps`

,
then the chain passes the halfwidth test. Otherwise, the chain fails
the halfwidth test and must be updated for more iterations until
sufficient accuracy is obtained. In order to avoid problems caused by
sequential testing, the test should not be repeated too frequently.
Heidelberger and Welch (1981) suggest increasing the run length by a
factor I > 1.5, each time, so that estimate has the same, reasonably
large, proportion of new data.

The Heidelberger and Welch MCMC convergence diagnostic conducts multiple hypothesis tests. The number of potentially wrong results increases with the number of non-independent hypothesis tests conducted.

The `Heidelberger.Diagnostic`

is a univariate diagnostic that is
usually applied to each marginal posterior distribution. A
multivariate form is not included. By chance alone due to multiple
independent tests, 5% of the marginal posterior distributions should
appear non-stationary when stationarity exists. Assessing multivariate
convergence is difficult.

Heidelberger, P. and Welch, P.D. (1981). "A Spectral Method for
Confidence Interval Generation and Run Length Control in Simulations".
*Comm. ACM.*, 24, p. 233--245.

Heidelberger, P. and Welch, P.D. (1983). "Simulation Run Length Control
in the Presence of an Initial Transient". *Opns Res.*, 31,
p. 1109--1144.

Schruben, L.W. (1982). "Detecting Initialization Bias in Simulation
Experiments". *Opns. Res.*, 30, p. 569--590.

`burnin`

,
`is.stationary`

,
`LaplacesDemon`

, and
`print.heidelberger`

.

```
# NOT RUN {
#library(LaplacesDemon)
###After updating with LaplacesDemon, do:
#hd <- Heidelberger.Diagnostic(Fit)
#print(hd)
# }
```

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