SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Next, we consider the case of iid data and investigate the RF stability by averaging out the randomness in data. Note that  $Z_{(n)}$  and  $\delta$  are random and depend on the data distribution, while we are interested in a probability bound for  $|\mathsf{RF}_B(X) - \mathsf{RF}_B^{\backslash i}(X)|$  greater than a deterministic quantity, which is only a function of  $B$  and  $n$ . In this finite-  $B$  case, the stability of  $\mathsf{RF}_B$  cannot be directly obtained from its conditional stability as in the derandomized situation.
Theorem 8 (Stability of finite- $B$  random forests). Assume training points in set  $D$  and the test point  $(X,Y)$  are iid, drawn from a fixed distribution. For the random forest predictor  $\mathsf{RF}_B$  consisting of  $B$  trees and trained on  $D$ , we have
$$
\mathbb {P} _ {D, X, \boldsymbol {\xi}, r} \left(\left| \mathrm {R F} _ {B} (X) - \mathrm {R F} _ {B} ^ {\backslash i} (X) \right| > \varepsilon_ {n, B}\right) \leqslant \nu_ {n, B}, \tag {10}
$$
where  $\varepsilon_{n,B} = \sum_{i=1}^{3}\varepsilon_{i}$ , and  $\nu_{n,B} = \sum_{i=1}^{3}\nu_{i}$ . The pair of  $(\varepsilon_{2},\nu_{2}/\lambda)$  satisfies the derandomized RF stability condition (7) with  $\lambda > 1$ . Moreover,  $\varepsilon_{1} = \varepsilon_{3} = \sqrt{2\lambda\mathbb{E}[Z_{(n)}^{2}]\ln\left(\frac{1}{\nu_{2}}\right)/B}$ ,  $\nu_{1} = 2\nu_{2} + 2\mathbb{P}(Z_{(n)}^{2} > \lambda\mathbb{E}[Z_{(n)}^{2}])$ , and  $\nu_{3} = g(p,\nu_{2},B) + 2\mathbb{P}(Z_{(n)}^{2} > \lambda\mathbb{E}[Z_{(n)}^{2}])$ .
On a high level, the establishment of this theorem relies on two observations: (i) the stability of the derandomized RF, so that the difference  $|\mathsf{rf}(X) - \mathsf{rf}^{\backslash i}(X)|$  is controlled, and (ii) the concentration of measure, so that the differences  $|\mathsf{RF}_B(X) - \mathsf{rf}(X)|$  and  $|\mathsf{RF}_B^{\backslash i}(X) - \mathsf{rf}^{\backslash i}(X)|$  are controlled. In order to make full sense of the word "stability," it is desirable that  $\varepsilon_{n,B}$  and  $\nu_{n,B}$  can converge to 0. It is known that  $\mathbb{E}[Y^2] < \infty$  suffices to ensure  $\mathbb{E}[Z_{(n)}^2] = o(n)$  [15, 13], and hence the stability of the derandomized RF. Now in the finite- $B$  case, we need an additional distributional assumption to control the tail probability  $\mathbb{P}(Z_{(n)}^2 > \lambda \mathbb{E}[Z_{(n)}^2])$ . It turns out that for typical light-tailed  $Y^2$ , such a tail probability will converge to 0 as  $n \to \infty$ . Technically, we can assume  $Y^2$  to be sub-gamma [8]. Note that bounded and sub-Gaussian random variables are sub-gamma. Hence the sub-gamma assumption is not strong and can be satisfied by distributions underlying many real datasets.
Definition 4 (Sub-gamma random variables [8]). A random variable  $W$  is said to be sub-gamma (on the right tail) with parameters  $(\sigma^2, c)$  where  $c \geqslant 0$ , if  $\ln \mathbb{E}[e^{s(W - \mathbb{E}[W])}] \leqslant \frac{s^2 \sigma^2}{2(1 - cs)}$  for all  $s \in (0, 1/c)$ .
Figure 1: Left: Density plots of the  $\log_{10}$  absolute difference  $|\mathsf{RF}_B(X) - \mathsf{RF}_B^{\backslash i}(X)|$  for 3000 OOB predictors  $\mathsf{RF}_B^{\backslash i}$  on 1000 test points. We let  $B = 1000$ . The RF stability (10) seems to persist, even though  $Y$  follows the (heavy-tailed) standard Cauchy distribution. Numerically, we set  $\hat{\nu}_{n,B} = 0.05$  and calculated the maximum of the 0.95 quantile of the 3000 empirical distributions to have  $\hat{\varepsilon}_{n,B} = 0.237$ . Right: Density plots of 1000  $\log_{10}$  absolute prediction errors  $|Y - \mathsf{RF}_B(X)|$  and of 3000  $\log_{10}$  absolute OOB errors  $|Y_i - \mathsf{RF}_B^{\backslash i}(X_i)|$ . The similarity between the plots supports the idea that the OOB errors can be used to construct PIs.
Lemma 1. Suppose  $Y^2$  is sub-gamma with parameters  $(\sigma^2, c)$  with  $c > 0$ , and  $\mathbb{E}[Z_{(n)}^2] \sim a \ln n$  with  $a \leqslant c$ . For  $\lambda > c / a$ , we have  $\lim_{n \to \infty} \mathbb{P}(Z_{(n)}^2 > \lambda \mathbb{E}[Z_{(n)}^2]) = 0$ .
Remark 5. We have set  $c > 0$  above. If  $c = 0$ , then  $Y^2$  is in fact sub-Gaussian, and the tail probability can be controlled similarly. If  $Y^2$  is upper bounded by some constant  $M^2$ , the stability analysis is even simpler, and there is no need to consider the tail probability at all, as we can use  $M^2$  in place of  $Z_{(n)}^2$  in the conditional stability of the RF and then take expectation with respect to data.
Example 1. Consider  $Y^2 \sim \mathrm{Exp}(1)$ , the exponential distribution with scale parameter 1. It is known that  $Y^2$  is sub-gamma with  $(\sigma^2, c) = (1, 1)$  [8], and  $\mathbb{E}[Z_{(n)}^2] = \sum_{i=1}^{n} \frac{1}{i} \equiv H_n$  with  $H_n \in (\gamma + \ln n, \gamma + \ln (n + 1))$ , where  $\gamma \approx 0.577$  is Euler's constant. Hence  $H_n = \ln n + o(\ln n)$ , and a straightforward calculation reveals that  $\lim_{n \to \infty} \mathbb{P}(Z_{(n)}^2 > \lambda \mathbb{E}[Z_{(n)}^2]) = 0$  as long as  $\lambda > 1$ .
From such results, one can see that the vanishing tail probability is not a stringent condition. By taking this additional assumption, it is indeed possible that both  $\varepsilon_{n,B}$  and  $\nu_{n,B}$  converge to 0.
Corollary 2. For the same setting as in Theorem 8, suppose  $Y^2$  is sub-gamma with parameters  $(\sigma^2, c)$  with  $c > 0$  and  $\mathbb{E}[Z_{(n)}^2] \sim a \ln n$  with  $a \leqslant c$ . Let  $\lambda > c / a$  be a fixed number, and let  $B$  depend on  $n$ . Then for  $\varepsilon_2$  that satisfies both  $\varepsilon_2 = \omega(\sqrt{\ln n / n})$  and  $\varepsilon_2 = o(1)$ , and  $B = \Omega(\ln^2 n)$ , we have  $\lim_{n \to \infty} \varepsilon_{n,B} = \lim_{n \to \infty} \nu_{n,B} = 0$ .
It is worth noting that there are multiple ways to let  $\varepsilon_{n,B}$  and  $\nu_{n,B}$  approach 0, as the dependence of  $\varepsilon_2$ ,  $B$ , and even  $\lambda$  on  $n$  can all be manipulated. The point is that, theoretically, even the greedy RF can be stable with vanishing parameters. In practice, however, the stability of  $\mathrm{RF}_B$  seems to hold in broader situations where both the moment and tail assumptions on  $Y^2$  can be relaxed.
3.4 Stability in practice and limitations of theory
We created a virtual dataset consisting of  $n = 4000$  points. We let  $Y$  be a standard Cauchy random variable, which is even without a well-defined mean. The feature vector  $X \in \mathbb{R}^3$  is determined as  $X = [0.5Y + \sin(Y), Y^2 - 0.2Y^3, \mathbb{I}\{Y > 0\} + \zeta]^T$  where  $\zeta$  is a standard normal random variable. We used 3000 of the points for training and 1000 of them as test points. Using the randomForest package with default setting (except letting  $B = 1000$ ), we had an output RF predictor  $\mathsf{RF}_B$ . We also aggregated corresponding tree predictors to have 3000 OOB predictors  $\mathsf{RF}_{B}^{i}$ . For each  $i \in [3000]$ , we calculated the absolute difference  $|\mathsf{RF}_B(X) - \mathsf{RF}_{B}^{i}(X)|$  on 1000 test points to come up with a
density plot for such a difference, shown in Fig. 1. We also calculated 1000 absolute prediction errors  $|Y - \mathsf{RF}_B(X)|$  that are incurred by  $\mathsf{RF}_B$  on test points, and 3000 OOB errors  $|Y_i - \mathsf{RF}_B^{\backslash i}(X_i)|$ , each incurred by an OOB predictor  $\mathsf{RF}_B^{\backslash i}$  on its OOB point  $(X_i, Y_i)$ . The computation can be done within a few minutes on a laptop. The density plots of these two kinds of errors are also shown in Fig. 1. This example shows that the RF stability can be present beyond the realm guaranteed by the light-tail assumption. As mentioned above, this is because the bound (2) can be conservative when  $n$  is large. We hope our results can inspire future study towards a more informative bound. Also, the similarity between the prediction error and the OOB error in this heavy-tail case indicates that the RF-based PIs analyzed below can find more applications in practice than justified by the current theory.
4 Random-forest prediction intervals
4.1 Comparison with related methods
With the stability property of the RF, it is possible to construct PIs with finite-sample guaranteed coverage. Recent years have witnessed the development of distribution-free predictive inference [1] with the full [33, 27], split [24, 31, 20], and jackknife+ [3, 32] conformal prediction methods being three milestones. The full conformal method is computationally prohibitive when used in practice. The split method greatly reduces the computational cost but fails to thoroughly extract the available information of training data. The jackknife+  $(\mathbf{J}+)$  method maximizes the usage of data at a computational cost in between those of full and split methods. In [18], jackknife+-after-bootstrap  $(\mathbf{J} + \mathbf{aB})$  was proposed for bagged algorithms to achieve the same goal as in  $\mathrm{J + }$ , while the training cost can be further reduced. However, the number of bags  $B$  is required to be a Binomial random variable, which might seem unnatural. It turns out that by further imposing the assumption of ensemble stability (which is essentially the concentration of resampling measure),  $\mathrm{J + aB}$  can still have guaranteed coverage with a fixed  $B$ . Ensemble stability is defined for bagged algorithms. It measures how typical a bootstrap sample is, and is different from the algorithmic stability that quantifies the influence of removing one training point. If algorithmic stability is also imposed, then not only  $\mathrm{J + aB}$ , but also jackknife can provide guaranteed coverage, which is otherwise impossible [29, 3].
Conceptually, the J+ approach and its variants under stability conditions are particularly relevant to this work. As the stability we establish for the RF contains both ensemble and algorithmic components, we will generally refer to the J+aB method with both ensemble and algorithmic stability as J+aBS and the jackknife method with algorithmic stability as JS. Our method might be best described as "jackknife-after-bootstrap-with-stability (JaBS)" tailored for the RF, which is different from both JS and J+aBS. Our method requires the least effort of computing as only one output predictor is needed, while all others require at least  $n$  output predictors.
There also exist RF-based PIs [17, 35] that are essentially of the jackknife-after-bootstrap (JaB) type and almost identical to ours practically when  $\varepsilon$  is small and  $n$  equals the size of a typical dataset. However, without stability, there is, in general, no guarantee for the coverage of such PIs, although the asymptotic coverage  $1 - \alpha$  can be established based on strong assumptions [35]. We take advantage of the stability of the RF algorithm to establish the lower bound of coverage in Theorem 9 below. An upper bound is established in Theorem 10 with an additional mild assumption. We also propose a weaker assumption for asymptotic coverage in Theorem 11.
We compare these relevant methods to ours in Table 1 and Table 2, where the RF is set as the working algorithm for all methods and  $(\varepsilon, \nu)$  is a general pair of stability parameters. We define  $q_{n,\alpha}\{R_i\}$ ,  $q_{n,\alpha}^{+}\{R_i\}$ ,  $q_{n,\alpha}^{-}\{R_i\}$ , and  $q_{n,\alpha}'\{R_i\}$  as follows. Given  $\{a_1, \ldots, a_n\}$ ,
$$
q _ {n, \alpha} \left\{a _ {i} \right\} = q _ {n, \alpha} ^ {+} \left\{a _ {i} \right\} \equiv \text {t h e} [ (1 - \alpha) (n + 1) ] - \text {t h s m a l l e s t v a l u e o f} \left\{a _ {1}, \dots , a _ {n} \right\},
$$
$$
q _ {n, \alpha} ^ {\prime} \left\{a _ {i} \right\} \equiv \text {t h e} [ (1 - \alpha) n ] - \text {t h s m a l l e s t v a l u e o f} \left\{a _ {1}, \dots , a _ {n} \right\},
$$
$$
q _ {n, \alpha} ^ {-} \left\{a _ {i} \right\} \equiv \text {t h e} [ \alpha (n + 1) ] - \text {t h s m a l l e s t v a l u e o f} \left\{a _ {1}, \dots , a _ {n} \right\},
$$
where  $\lfloor \cdot \rfloor$  is the floor function. Let  $R_{i}^{\mathrm{LOO}} = |Y_{i} - \mathsf{RF}_{B}^{-i}(X_{i})|$  be the LOO error, where  $\mathsf{RF}_B^{-i}$  is trained without the ith training point, and by definition  $\mathsf{RF}_B^{-i}\neq \mathsf{RF}_B^{\backslash i}$ .
In Table 1, we list the corresponding PI constructed from each method and the output predictors of each method. The number of output predictors directly reflects the computational cost. It is
Table 1: Methods to construct prediction intervals using random forests: computational cost
Method	Output predictors	Prediction interval for future Y
J+ [3]	RF-Bi, i ∈ [n]	[q-n,α{RF-Bi(X) - RiLOO}, q+n,α{RF-Bi(X) + RiLOO}]
J+aB [18]	RFiB, i ∈ [n]	[q-n,α{RFiB(X) - Ri}, q+n,α{RFiB(X) + Ri}]
JS [3]	RF and RF-Bi, i ∈ [n]	RF(B)(X) ± qn,α{RILOO + ε}
J+aBS [18]	RFiB, i ∈ [n]	[q-n,α{RFiB(X) - Ri} - ε, q+n,α{RFiB(X) + Ri} + ε]
JaB	RFB	RF(B)(X) ± qn,α{Ri} [17]
RF(B)(X) ± q′n,α{Ri} [35]
Ours (JaBS)	RFB	RF(B)(X) ± qn,α{Ri + ε} (Theorem 9)
RF(B)(X) ± qn,α{Ri - ε} (Theorem 10)
RF(B)(X) ± qn,α{Ri} (Theorem 11)
Table 2: Methods to construct prediction intervals using random forests: theoretical coverage
Method	Theoretical coverage	Additional conditions
J+ [3]	≥ 1 - 2α	None
J+aB [18]	≥ 1 - 2α	Binomial B
JS [3]	≥ 1 - α - O(√ν)	Stability (algorithmic)
J+aBS [18]	≥ 1 - α - O(√ν)	Stability (ensemble + algorithmic)
JaB	No guarantee [17]	-
→ 1 - α [35]	Strong (additive model, consistency of RF predictor)
Ours (JaBS)	≥ 1 - α - O(√ν)	Stability (Theorem 9)
≤ 1 - α + 1/n+1 + O(√ν)	+ Distinct residuals (Theorem 10)
→ 1 - α	+ Uniformly equicontinuous CDF of |Y - RF(B(X)| and vanishing ε, ν (Theorem 11)
worth noting that acquiring the LOO predictor  $\mathsf{RF}_B^{-i}$  needs substantial computation. In packages like randomForest, aggregating tree predictors to obtain the OOB predictor  $\mathsf{RF}_B^{\backslash i}$  also needs extra computation. However, the predicted value  $\mathsf{RF}_B^{\backslash i}(X_i)$  can be obtained immediately by calling the predict() function. The fact that the value of  $\mathsf{RF}_B^{\backslash i}(X)$  on a test point is not needed further reduces the computational cost of JaB and our method, which only need one output RF predictor, and are more favorable computationally.
In Table 2, we list the coverage of the PI constructed from each method, as well as the additional conditions (beyond iid data) needed to achieve the coverage. Note that  $\mathrm{J + }$  does not require any additional conditions to achieve the coverage lower bound  $1 - 2\alpha$  , but  $\mathrm{J + aB}$  requires that the number of trees  $B$  be a Binomial random variable. For JS, J+aBS, and our method, stability is needed to achieve the coverage lower bound  $1 - \alpha -O(\sqrt{\nu})$  . With additional mild assumptions, the coverage upper bound and asymptotic coverage of our method can be established. However, there is no guarantee of coverage for JaB without strong assumptions.
In summary, our theoretical work provides a series of coverage guarantees to a computationally feasible method for constructing PIs based on the RF algorithm. In the following, we will establish the lower and upper bound of coverage, as well as the asymptotic coverage.
4.2 Non-asymptotic coverage guarantees
Theorem 9 (Coverage lower bound). Suppose the RF predictor  $\mathsf{RF}_B$  satisfies the stability condition as in Theorem 8. Then we have for a test point  $(X,Y)$  that
$$
\mathbb {P} \left(Y \in \mathrm {R F} _ {B} (X) \pm q _ {n, \alpha} \left\{R _ {i} + \varepsilon_ {n, B} \right\}\right) \geqslant 1 - \alpha - \nu_ {1} - 2 \sqrt {\nu_ {2}} - 2 \sqrt {\nu_ {3}}. \tag {11}
$$
This result is established by starting from the analysis of an imaginary extended dataset  $\overline{D} = D \cup \{(X, Y)\}$ , where the test point is assumed to be known. We denote  $(X, Y)$  as  $(X_{n+1}, Y_{n+1})$  for convenience. For all points in  $\overline{D}$ , consider the derandomized RF predictor  $\widetilde{\mathrm{rf}}^{\backslash i}$  that is built on  $n$  data
points without the  $i$ th point in  $\overline{D}$ ,  $i \in [n+1]$ . One can then define the OOB error  $\widetilde{r}_i \equiv |Y_i - \widetilde{\mathsf{rf}}^{\backslash i}|$ . Since all data are iid, we have that  $\mathbb{P}(\widetilde{r}_{n+1} \leqslant q_{n,\alpha}\{\widetilde{r}_i\}) \geqslant 1 - \alpha$ , where  $q_{n,\alpha}\{\widetilde{r}_i\}$  is the  $[(1 - \alpha)(n+1)]$ -th smallest value of  $\{\widetilde{r}_1, \ldots, \widetilde{r}_n\}$ . Next, notice  $\widetilde{r}_{n+1} = |Y_{n+1} - \widetilde{\mathsf{rf}}^{(n+1)}(X_{n+1})| = |Y_{n+1} - \mathsf{rf}(X_{n+1})|$  by the definitions of  $\widetilde{\mathsf{rf}}^{(n+1)}$  and  $\mathsf{rf}$ . By concentration of measure,  $\mathsf{rf}(X_{n+1})$  can be approximated by  $\mathsf{RF}_B(X_{n+1})$ , and thus  $\widetilde{r}_{n+1}$  can be roughly replaced with  $|Y_{n+1} - \mathsf{RF}_B(X_{n+1})|$ , which is desired. Then by stability of  $\mathsf{rf}$ ,  $\{\widetilde{r}_i\}$  can be approximated by  $\{r_i \equiv |Y_i - \mathsf{rf}^{\backslash i}(X_i)|\}$ . Although  $\{r_i\}$  is still unavailable in practice, by applying the idea of concentration of measure again,  $\{r_i\}$  can be further approximated by  $\{R_i\}$ , which is accessible given  $D$ . Eventually, we can bound  $|Y_{n+1} - \mathsf{RF}_B(X_{n+1})|$  in terms of  $\{R_i\}$ . The approximations are accounted for by the stability parameters in Theorem 8.
If we further assume that there are no ties among  $\{\widetilde{r}_i\}$ ,  $i \in [n + 1]$ , a typical case when  $Y$  is continuous, then we can also establish the upper bound of coverage.
Theorem 10 (Coverage upper bound). Suppose there are no ties in  $\{\widetilde{r}_i\}$ ,  $i \in [n + 1]$ , and the RF predictor  $\mathsf{RF}_B$  satisfies the stability condition as in Theorem 8. Then
$$
\mathbb {P} \left(Y \in \mathrm {R F} _ {B} (X) \pm q _ {n, \alpha} \left\{R _ {i} - \varepsilon_ {n, B} \right\}\right) \leqslant 1 - \alpha + \frac {1}{n + 1} + \nu_ {1} + 2 \sqrt {\nu_ {2}} + 2 \sqrt {\nu_ {3}}. \tag {12}
$$
The upper bound can be established because if there are no ties among  $\widetilde{r}_1, \ldots, \widetilde{r}_{n+1}$ , then  $\mathbb{P}(\widetilde{r}_{n+1} \leqslant q_{n,\alpha} \{\widetilde{r}_i\}) \leqslant 1 - \alpha + \frac{1}{n+1}$ . The apparent symmetry between the lower and upper bound originates from the fact that they both are established by using the RF stability once and the concentration of measure twice. Note that this idea can be applied to JS intervals for an arbitrary stable algorithm in exactly the same way, providing a complement to the lower bound for JS intervals established in [3].
Corollary 3 (Coverage upper bound for jackknife-with-stability intervals). Let  $\hat{f}$  be a predictor trained on  $n$  iid data points and  $\hat{f}^{-i}$  be the LOO predictor without the  $i$ th point. Suppose  $\hat{f}$  is stable with  $\mathbb{P}(|\hat{f}(X) - \hat{f}^{-i}(X)| > \varepsilon) \leqslant \nu$ , and the LOO errors are distinct on the extended training set that includes an iid test point  $(X,Y)$ . Then we have  $\mathbb{P}\left(|Y - \hat{f}(X)| \leqslant q_{n,\alpha}\{r_i - \varepsilon\}\right) \leqslant 1 - \alpha + \frac{1}{n+1} + 2\sqrt{\nu}$ , where  $r_i$  are the LOO errors on the original training set.
4.3 Asymptotic coverage guarantee
As shown above, the stability parameters  $(\varepsilon_{n,B},\nu_{n,B})$  can vanish when  $n\to \infty$ . It is reasonable to expect that  $\mathbb{P}(Y\in \mathsf{RF}_B(X)\pm q_{n,\alpha}\{R_i\})\to 1 - \alpha$  in this limit, as is consistent with numerous empirical observations [17, 35]. However, to achieve this goal, it seems that more assumptions are unavoidable. In [35], the guaranteed coverage of the JaB method is established by assuming that  $\mathsf{RF}_B(X)$  converges to some  $f_0(X)$  in probability as  $n\to \infty$ , where  $f_{0}$  is the true regression function of an additive model that generates the data. We show that this can be done under weaker conditions.
Theorem 11 (Asymptotic coverage). Denote  $F_{n}$  as the CDF of  $|Y - \mathsf{RF}_B(X)|$ . Suppose  $\{F_n\}_{n\geqslant n_0}$  is uniformly equicontinuous for some  $n_0$ . Then  $\mathbb{P}(Y\in \mathsf{RF}_B(X)\pm q_{n,\alpha}\{R_i\})\to 1 - \alpha$  as  $n\to \infty$  when conditions in Theorem 9, Theorem 10, and Corollary 2 are satisfied.
Remark 6. Intuitively, using errors from  $\mathsf{RF}_B^{\backslash i}$  that are trained on  $n - 1$  points to approximate those from  $\mathsf{RF}_B$ , trained on  $n$  points, we only need this approximation to be exact asymptotically. There is no need for  $\mathsf{RF}_B$  itself to converge to anything. This is one major conceptual difference between our work and [35], and it is in this sense that our assumption is weaker. Practically, this kind of PI is recommended as it does not involve  $(\varepsilon_{n,B},\nu_{n,B})$ , and has great performance on numerous datasets.
In this work, for the first time, we theoretically establish the stability property of the greedy version of random forests, which is implemented in popular packages. The theoretical guarantee is based on a light-tail assumption of the marginal distribution of the squared response  $Y^2$ . However, numerical evidence suggests that this stability could persist in much broader situations. Based on the stability property and some mild conditions, we also establish finite-sample lower and upper bounds of coverage, as well as the exact coverage asymptotically, for prediction intervals constructed from the out-of-bag error of random forests, justifying random forests as an appealing method to provide both point and interval prediction simultaneously.
Much of the work was completed while Yan Wang was a PhD student in the Department of Statistics at Iowa State University. This research was supported in part by the US National Science Foundation under grant HDR:TRIPODS 19-34884.