**Table 1.** FLS Data—prior and posterior moments of model size Properties of the chain and the best model

|||_m_|= 7|||_m_=_k/_|2 = 20_._5||
|---|---|---|---|---|---|---|---|---|
||fx|ed_θ_|rand|om_θ_|fxe|d_θ_|rand|om_θ_|
||_g_ = 1_/n_|_g_ = 1_/k_<sup>2</sup>|_g_ = 1_/n_|_g_ = 1_/k_<sup>2</sup>|_g_ = 1_/n_|_g_ = 1_/k_<sup>2</sup>|_g_ = 1_/n_|_g_ = 1_/k_<sup>2</sup>|
|Prior mean|7|7|7|7|20.5|20.5|20.5|20.5|
|Prior st. dev.|2.41|2.41|6.30|6.30|3.20|3.20|12.12|12.12|
|Posterior mean|9.17|6.29|10.76|5.73|19.84|9.91|12.87|6.03|
|Posterior st. dev.|1.54|1.30|2.67|1.65|2.57|1.63|4.43|1.64|
|# Models visited|141,980|20,003|329,694|19,275|522,637|150,255|436,602|23,558|
|# Models covering 50%|1,461|40|3,569|27|9,348|1,598|5,489|36|
|Post. Prob. best model|0.91%|6.22%|0.65%|5.61%|0_._21%|1.24%|0.54%|5.21%|
|_kj_ for best model|9|6|10|5|20|10|10|6|
|Corr. visits and PO|0.992|0.999|0.971|0.999|0.748|0.992|0.922|0.998|
|Prob. mass visited|70.7%|95.2%|39.6%|95.1%|18.4%|69.7%|27.2%|93.7%|

that the prior on model size is very concentrated for the fixed _θ_ case, and the relative robustness with respect to the choice of _m_ noted for random _θ_ in Subsections 3.2 and 3.3. Also, it appears that the effects of prior choices are much more pronounced for the prior with _g_ = 1 _/n_ . In fact, if we adopt _g_ = 1 _/k_<sup>2</sup> with a hierarchical prior on _θ_ , the value chosen for _m_ has very little impact, in marked contrast with the other prior settings.

From Figure 5 we also deduce that the posterior of _W_ can display bimodality if data and prior information are in conflict. This happens when _m_ = 20 _._ 5 and we choose _g_ = 1 _/n_ for the case with random _θ_ . In the corresponding fixed _θ_ case the prior is actually so concentrated that it dominates and the posterior is mostly determined by the prior.

Table 1 also records some key properties of the chain: it is clear that the choice of _g_ = 1 _/n_ always leads to the sampler visiting many more models, indicating that the posterior mass is more spread out over the model space. This is obvious when considering how many of the higher-probability models we need to cover 50% of the probability mass. This varies dramatically with the choice of _g_ , especially for random _θ_ (where we need more than 100 times as many models for _g_ = 1 _/n_ ). If we adopt _g_ = 1 _/n_ , many larger models are visited and the posterior mass assigned to the best model is much smaller than with _g_ = 1 _/k_<sup>2</sup> (roughly by a factor 8). In addition, the best model is much larger for _g_ = 1 _/n_ . All this is in keeping with the smaller penalty for increasing model size that is implied by adopting _g_ = 1 _/n_ (see Subsection 3.3). An important practical consequence of this is that the chain should run longer before convergence is ensured. Table 1 also presents the correlation between model visit frequencies and probabilities computed on the basis of the exact posterior odds of the visited models, which is in excess of 0.99 (indicating excellent convergence) for all cases with _g_ = 1 _/k_<sup>2</sup> . For _g_ = 1 _/n_ , where the model probability is spread more evenly over _M_ , the evidence in favor of convergence is slightly less convincing. For the case with _m_ = 20 _._ 5 and fixed _θ_ we would ideally recommend a longer run.<sup>6</sup> However, even in this case, multiple runs

> 6 It is important for our purposes in this section that run lengths are identical, to ensure comparability of the properties of

12