Description

Takes posterior draws from Butts' bayesian network accuracy/estimation model for multiple participant/observers (conditional on observed data and priors), using a Gibbs sampler.

Usage

Arguments

Details

The bbnam models a set of network data as reflecting a series of (noisy) observations by a set of participants/observers regarding an uncertain criterion structure. Each observer is assumed to send false positives (i.e., reporting a tie when none exists in the criterion structure) with probability e^+, and false negatives (i.e., reporting that no tie exists when one does in fact exist in the criterion structure) with probability e^-. The criterion network itself is taken to be a Bernoulli (di)graph. Note that the present model includes three variants:

1. Fixed error probabilities: Each edge is associated with a known pair of false negative/false positive error probabilities (provided by the researcher). In this case, the posterior for the criterion graph takes the form of a matrix of Bernoulli parameters, with each edge being independent conditional on the parameter matrix.

2. Pooled error probabilities: One pair of (uncertain) false negative/false positive error probabilities is assumed to hold for all observations. Here, we assume that the researcher's prior information regarding these parameters can be expressed as a pair of Beta distributions, with the additional assumption of independence in the prior distribution. Note that error rates and edge probabilities are not independent in the joint posterior, but the posterior marginals take the form of Beta mixtures and Bernoulli parameters, respectively.

3. Per observer (“actor”) error probabilities: One pair of (uncertain) false negative/false positive error probabilities is assumed to hold for each observation slice. Again, we assume that prior knowledge can be expressed in terms of independent Beta distributions (along with the Bernoulli prior for the criterion graph) and the resulting posterior marginals are Beta mixtures and a Bernoulli graph. (Again, it should be noted that independence in the priors does not imply independence in the joint posterior!)

By default, the bbnam routine returns (approximately) independent draws from the joint posterior distribution, each draw yielding one realization of the criterion network and one collection of accuracy parameters (i.e., probabilities of false positives/negatives). This is accomplished via a Gibbs sampler in the case of the pooled/actor model, and by direct sampling for the fixed probability model. In the special case of the fixed probability model, it is also possible to obtain directly the posterior for the criterion graph (expressed as a matrix of Bernoulli parameters); this can be controlled by the outmode parameter.

As noted, the taking of posterior draws in the nontrivial case is accomplished via a Markov Chain Monte Carlo method, in particular the Gibbs sampler; the high dimensionality of the problem (O(n^2+2n)) tends to preclude more direct approaches. At present, chain burn-in is determined ex ante on a more or less arbitrary basis by specification of the burntime parameter. Eventually, a more systematic approach will be utilized. Note that insufficient burn-in will result in inaccurate posterior sampling, so it's not wise to skimp on burn time where otherwise possible. Similarly, it is wise to employ more than one Markov Chain (set by reps), since it is possible for trajectories to become “trapped” in metastable regions of the state space. Number of draws per chain being equal, more replications are usually better than few; consult Gelman et al. for details. A useful measure of chain convergence, Gelman and Rubin's potential scale reduction (√{\hat{R}}), can be computed using the compute.sqrtrhat parameter. The potential scale reduction measure is an ANOVA-like comparison of within-chain versus between-chain variance; it approaches 1 (from above) as the chain converges, and longer burn-in times are strongly recommended for chains with scale reductions in excess of 1.2 or thereabouts.

Finally, a cautionary concerning prior distributions: it is important that the specified priors actually reflect the prior knowledge of the researcher; otherwise, the posterior will be inadequately informed. In particular, note that an uninformative prior on the accuracy probabilities implies that it is a priori equally probable that any given actor's observations will be informative or negatively informative (i.e., that i observing j sending a tie to k reduces Pr(j->k)). This is a highly unrealistic assumption, and it will tend to produce posteriors which are bimodal (one mode being related to the “informative” solution, the other to the “negatively informative” solution). Currently, the default error parameter prior is Beta(1,11), which is both diffuse and which renders negatively informative observers extremely improbable (i.e., on the order of 1e-6). Another plausible but still fairly diffuse prior would be Beta(3,5), which reduces the prior probability of an actor's being negatively informative to 0.16, and the prior probability of any given actor's being more than 50% likely to make a particular error (on average) to around 0.22. (This prior also puts substantial mass near the 0.5 point, which would seem consonant with the BKS studies.) For network priors, a reasonable starting point can often be derived by considering the expected mean degree of the criterion graph: if d represents the user's prior expectation for the mean degree, then d/(N-1) is a natural starting point for the cell values of nprior. Butts (2003) discusses a number of issues related to choice of priors for the bbnam model, and users should consult this reference if matters are unclear before defaulting to the uninformative solution.

Value

An object of class bbnam, containing the posterior draws. The components of the output are as follows:

Note

As indicated, the posterior draws are conditional on the observed data, and hence on the data collection mechanism if the collection design is non-ignorable. Complete data (e.g., a CSS) and random tie samples are examples of ignorable designs; see Gelman et al. for more information concerning ignorability.

Author(s)

Carter T. Butts buttsc@uci.edu

References

Butts, C. T. (2003). “Network Inference, Error, and Informant (In)Accuracy: A Bayesian Approach.” Social Networks, 25(2), 103-140.

Gelman, A.; Carlin, J.B.; Stern, H.S.; and Rubin, D.B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

Gelman, A., and Rubin, D.B. (1992). “Inference from Iterative Simulation Using Multiple Sequences.” Statistical Science, 7, 457-511.

Krackhardt, D. (1987). “Cognitive Social Structures.” Social Networks, 9, 109-134.

See Also

npostpred, event2dichot, bbnam.bf

Examples