where _ε_ is a vector of iid standard normal random variables and we take _σ_ = 2 _._ 5. In Model 1 we use _n_ = 50 _,_ 100 and for Model 2 we use _n_ = 50 _,_ 100 _,_ 1000 _,_ 10 _,_ 000 in order to illustrate consistency.

The remaining model structure used in the simulations uses regressors with pairwise correlations of 0.5, generated as _r_ ( _i_ ) = _r_ (<sup>_⋆_</sup> _i_ )<sup>+</sup><sup>_e_where each</sup><sup>_r_</sup> (<sup>_⋆_</sup> _i_ )<sup>and</sup><sup>_e_are vectors of</sup><sup>_n_independent</sup> standard normal elements. After demeaning to obtain _Z_ we generate _n_ observations as Model 3: _y_ = _ιn_ +<sup>�7</sup> _h_ =1<sup>_z_(</sup><sup>_h_) +</sup><sup>_σε,_</sup> where now we use _σ_ = 2. Values of _k_ used are 20, 40 and 80, while we adopt _n_ = 50 and 100.<sup>11</sup> We analyse 100 simulated data sets for each model and value of _k_ and _n_ mentioned above. The prior on model space is constructed with a random _θ_ as in (5) with a prior mean model size _m_ = _k/_ 2. The different priors for _g_ discussed above have been used, as well as two fixed values for _g_ ( _g_ = _n_ and _g_ = _k_<sup>2</sup> ). For the hyper- _g/n_ prior we take _a_ = 3.

Rather than provide exhaustive results from these simulations, we merely highlight the most important findings here. The different priors do lead to rather different posterior distributions for _g_ . Broadly speaking, the hyper- _g_ priors lead to the smallest median posterior median<sup>12</sup> for _g_ , while the benchmark prior with _c_ = 1 often leads to the largest. From Section 5, we know that larger _g_ induces a higher model size penalty ( _cpBF_ ) and this is immediately reflected in the results. Consider, in particular, the average number of regressors in the visited models. Tables 2 and 3 record the median over the 100 samples of the average model size. As expected, average model sizes are smaller for the priors leading to higher values of _g_ , in particular the benchmark prior with _c_ = 1 and the case with fixed _g_ = _k_<sup>2</sup> . On the other hand, the hyper- _g_ priors and the Bottolo-Richardson and horseshoe priors favour larger models. This is a serious drawback in the case of Model 2, where data are generated from the null model. Here the hyper- _g_ and horseshoe priors (and to a slightly lesser extent the Bottolo-Richardson prior) are choosing models that are far too large: marginal inclusion probabilities of all 15 regressors (none of which are used in the true model) are typically over 0.4 with these priors. In contrast, the probability of erroneously including the regressors tends to be under 1% for the benchmark priors with _c_ = 1 and _c_ = 0 _._ 1 and the case with _g_ = _k_<sup>2</sup> . The priors of Maruyama-George, Forte _et al._ and Zellner-Siow and the case with _g_ = _n_ typically lead to marginal inclusion probabilities under 2.5%.

Tables 2 and 3 also present the median posterior probability of the true model. These tend to be smallest for the hyper- _g_ priors throughout and for the Bottolo-Richardson prior in the case of Model 3. The highest probabilities on the true model are typically found

> 11 The situation with _n_ = 50 and _k_ = 80 means that we need to exclude models with _kj ≥ n −_ 1, as the posterior for such a model would no longer be well-defined. This is simply done by imposing a prior on model space that limits the model size to _n −_ 2. The sampler then rejects any proposed model for which _kj > n −_ 2. In the analysis of these simulated data, the sampler never even gets close to models that large.

> 12 Here and in the subsequent discussion we often focus on the median over the 100 samples we generated. We do not consider the median of the posterior means of _g_ , since the posterior mean of _g_ does not exist for all but the Bottolo-Richardson prior (see the discussion in Subsection 3.3).

19