Consort: Consort with Laplace's Demon In LaplacesDemon: Complete Environment for Bayesian Inference

Description

This may be used to consort with Laplace's Demon regarding an object of class `demonoid`. Laplace's Demon will offer suggestions.

Usage

 `1` ```Consort(object) ```

Arguments

 `object` This required argument is an object of class `demonoid`. For more information, see the `LaplacesDemon` function.

Details

First, `Consort` calls `print.demonoid`, which prints most of the components to the screen from the supplied object of class `demonoid`.

Second, Laplace's Demon considers a combination of five conditions when making the largest part of its suggestion. These conditions are: the algorithm, acceptance rate, MCSE, ESS, and stationarity. Other things are considered as well, such as the recommended thinning value is used to suggest a new number of iterations, how fast the algorithm is expected to be, and if the condition of diminishing adaptation (also called the vanishing adaptation condition) was met (for an adaptive algorithm). Diminishing adaptation occurs only when the absolute value of the proposed variances trends downward (toward zero) over the course of all adaptations. When an algorithm is adaptive and it does not have diminishing adaptations, the `Consort` function will suggest a different adaptive algorithm. The `Periodicity` argument is suggested to be set equal to the value of `Rec.Thinning`.

Appeasement applies only when all parameters are continuous.The `Hangartner.Diagnostic` should be considered for discrete parameters.

Appeasement Conditions

• Algorithm: The final algorithm must be non-adaptive, so that the Markov property holds. This is conservative. A user may have an adaptive (non-final) algorithm in which adaptations in the latest update are stationary, or no longer diminishing. Laplace's Demon is unaware of previous updates, and conservatively interprets this as failing to meet the condition of diminishing adaptation, when the output may be satisfactory. On the other hand, if the adaptive algorithm has essentially stopped adapting, and if there is a non-adaptive version, then the user should consider switching to the non-adaptive algorithm. User discretion is advised.

• Acceptance Rate: The acceptance rate is considered satisfactory if it is within the interval [15%,50%] for most algorithms. Some algorithms have different recommended intervals.

• MCSE: The Monte Carlo Standard Error (MCSE) is considered satisfactory for each target distribution if it is less than 6.27% of the standard deviation of the target distribution. This allows the true mean to be within 5% of the area under a Gaussian distribution around the estimated mean. The `MCSE` function is used. Toft et al. (2007) propose a stricter criterion of 5%. The criterion of 6.27% for this stopping rule is arbitrary, and may be too lenient or strict, depending on the needs of the user. Nonetheless, it has performed well, and this type of stopping rule has been observed to perform better than MCMC convergence diagnostics (Flegal et al., 2008).

• ESS: The effective sample size (ESS) is considered satisfactory for each target distribution if it is at least 100, which is usually enough to describe 95% probability intervals (see `p.interval` and `LPL.interval` for more information). The `ESS` function is used. When this criterion is unmet, the name of the worst mixing chain in Summary1 appears.

• Stationarity: Each target distribution is considered satisfactory if it is estimated to be stationary with the `BMK.Diagnostic` function.

Bear in mind that the MCSE, ESS, and stationarity criteria are all univariate measures applied to each marginal posterior distribution. Multivariate forms are not included. By chance alone due to multiple independent tests, 5% of these diagnostics should indicate non-convergence when 'convergence' exists. In contrast, even one non-convergent nuisance parameter is associated with non-convergence in all other parameters. Assessing convergence is difficult.

If all five conditions are satisfactory, then Laplace's Demon is appeased. Otherwise, Laplace's Demon will suggest and supply R code that is ready to be copy/pasted and executed.

To visualize the MCSE-based stopping rule, run the following code:

`x <- seq(from=-3, to=3, by=0.1);` `plot(x, dnorm(x,0,1), type="l");` `abline(v=-0.0627); abline(v=0.0627);` `abline(v=2*-0.0627, col="red"); abline(v=2*0.0627, col="red")`

The black vertical lines show the standard error, and the red vertical lines show the 95% interval.

If the user has an object of class `demonoid.hpc`, then the `Consort` function may be still be applied, but a particular chain in the object must be specified as a component in a list. For example, with an object called `Fit` and a goal of consorting over the second chain, the code would be: `Consort(Fit[[2]])`.

The Demonic Suggestion is usually very helpful, but should not be followed blindly. Do not let it replace critical thinking. For example, `Consort` may find that diminishing adaptation is unmet, and recommend a different algorithm. However, the user may be convinced that the current algorithm is best, and believe instead that MCMC found a local solution, and is leaving it to find the global solution, in which case adaptations may increase again. Diminishing adaptation may have occurred in a previous run, and is not found in the current run because adaptation is essentially finished. If either of these is true, then it may be best to ignore the newly suggested algorithm, and continue with the current algorithm. The suggested code may be helpful, but it is merely a suggestion.

If achieving the appeasement of Laplace's Demon is difficult, consider ignoring the MCSE criterion and terminate when all other criteria have been met, placing special emphasis on ESS.

Author(s)

Statisticat, LLC. software@bayesian-inference.com

References

Flegal, J.M., Haran, M., and Jones, G.L. (2008). "Markov chain Monte Carlo: Can We Trust the Third Significant Figure?". Statistical Science, 23, p. 250–260.

Toft, N., Innocent, G., Gettinby, G., and Reid, S. (2007). "Assessing the Convergence of Markov Chain Monte Carlo Methods: An Example from Evaluation of Diagnostic Tests in Absence of a Gold Standard". Preventive Veterinary Medicine, 79, p. 244–256.

`BMK.Diagnostic`, `ESS`, `Hangartner.Diagnostic`, `LaplacesDemon`, `LaplacesDemon.hpc`, `LPL.interval`, `MCSE`, and `p.interval`.