define a prior on $\Theta$ such that it places positive probability in any small neighborhood of $\Theta_0$, according to $d_\infty$. The prior defined in *Section 2.2* satisfies this condition. Consequently, the posterior distribution of $\Theta$ is weakly consistent due to [@schwartz1965bayes]. The proofs of *Proposition 2* and *Proposition 3* can be found in the *Supplementary Material*.

Simulation Experiments
======================

In this section we compare the performance of our FIN method with four other approaches: PIE [@wang2019penalized], RAMP [@HaoFengZhang2016], Family [@Haris2018] and HierNet [@Bien2013]. These methods are designed for inference on interactions in moderate to high dimensional settings. We generate $25$ and $50$ covariates in three ways: $$\begin{aligned}
&  X_i \sim N_p(0,\Lambda \Lambda^T +I_p), \quad \lambda_{i,j} \sim N(0,1), \quad && \text{(factor)} \\
&    X_i \sim N_p(0,W), \quad [W]_{i,j} = 0.8 |i - j|, \quad && \text{(linear)}\\
&    X_i \sim N_p(0,I_p). \quad && \text{(independent)}\end{aligned}$$ In the factor scenario we set the true number of factors equal to $7$ for $p=25$ and equal to $17$ when $p = 50$. FIN achieved similar performance when we chose a smaller number of latent factors. The average absolute correlation in the covariates is between $0.25$ and $0.3$ for the factor and linear scenarios when $p = 