# Efficient Estimation of Generative Models Using Tukey Depth
## 3 Tukey Depth  
To simplify notation, in this section, we avoid references to the parameters \(\theta\) of the probability distribution. For a point \(x\in\mathbb{R}^{p}\), let \(D(x,P)\) denote the depth function of \(x\) with respect to the probability distribution \(P\). A high value of the depth function refers to a central point of the probability distribution, and a low value of the depth function refers to an outlying point. A general depth function is defined by satisfying the natural requirements of affine invariance, maximality at the centre, monotonicity relative to the deepest point, and vanishing at infinity [41]. Given a set of points \(\mathcal{P}\) and a point \(x\) in \(k\)-dimensional space, the Tukey depth of \(x\) with respect to the set \(\mathcal{P}\) is the smallest number of points in any closed halfspace that contains \(x\). This leads us to the following definition. **Definition 1** (Tukey depth). : _Let \(\mathcal{S}\) refer to the set of all unit vectors in \(\mathbb{R}^{p}\). Tukey depth is the minimum probability mass carried by any closed halfspace containing the point_  
\[D(x,P)=\inf_{u\in\mathcal{S}}P\Big{(}u^{T}X\leq u^{T}x\Big{)}. \tag{2}\]  
Next, given \(\alpha>0\), we will define \(\alpha\)-depth region, contour, directional quantile, and directional quantile halfspace with respect to Tukey depth. **Definition 2** (\(\alpha\)-depth region, contour). : _Given \(\alpha>0\), the \(\alpha\)-depth region with respect to Tukey depth, denoted by \(R(\alpha)\), is defined as the set of points with depth at least \(\alpha\)_  
\[R(\alpha)=\{x\in\mathbb{R}^{p}\colon D(x,P)\geq\alpha\}. \tag{3}\]  
_The boundary of \(R(\alpha)\) is referred to as the \(\alpha\)-depth contour or Tukey depth contour (TDC)._  
Note that the \(\alpha\)-depth regions are closed, convex, and nested for increasing \(\alpha\). **Definition 3** (Directional quantile). : _For any unit directional vector \(u\in\mathcal{S}\), define the (\(\alpha\)-depth) directional quantile as_  
\[Q(\alpha,u^{T}X)=F_{u^{T}X}^{-1}(\alpha), \tag{4}\]  
_where \(F_{u^{T}X}^{-1}(x)\) refers to the inverse of the univariate cumulative distribution function of the projection of \(X\) on \(u\)._  
**Definition 4** (Directional quantile halfspace). : _The (\(\alpha\)-depth) directional quantile halfspace with respect to some directional vector \(u\in\mathcal{S}\) is defined as_  
\[H(\alpha,u)=\Big{\{}x\in\mathbb{R}^{p}\colon u^{T}x\geq Q(\alpha,u^{T}X)\Big{\}}, \tag{5}\]  
_which is bounded away from the origin at distance \(Q(\alpha,u^{T}X)\) by the hyperplane with normal vector \(u\)._  
Consequently, we always have \(P(X\in H(\alpha,u))=1-\alpha\) for any \(u\in\mathcal{S}\). We now finish this section by noting that the estimation procedures in this paper build on the following theorem from [8]. **Theorem 1**. : _The \(\alpha\)-depth region in (3) equals the directional quantile envelope_  
\[D(\alpha)=\bigcap_{u\in\mathcal{S}}H(\alpha,u). \tag{6}\]  
_Estimating \(\alpha\)-Depth Region from Observations_  
Let \(\mathbf{X}=X_{1},\ldots,X_{n}\) represent a random sample from the distribution of \(X\). Suppose that we want to use the random sample to estimate the \(\alpha\)-depth region for some \(\alpha>0\). In this paper, we suggest doing this by approximating Equation (6) using a finite number of directional vectors \(u_{i},\ i\in 1,\ldots,n_{u}\) from \(\mathcal{S}\). Furthermore, following (4), we also note that the directional quantile corresponding to each directional vector \(u_{i}\), denoted by \(\widehat{Q}_{i}(\alpha,\mathbf{X})\), can be approximated from the random sample \(\mathbf{X}\) by computing the \(\alpha\)-quantile of \(u_{i}^{T}X_{1},\ldots,u_{i}^{T}X_{n}\).