Title: Generative Regression with IQ-BART Show Chinese title

Title: 带有IQ-BART的生成回归

Abstract: Implicit Quantile BART (IQ-BART) posits a non-parametric Bayesian model on the conditional quantile function, acting as a model over a conditional model for $Y$ given $X$. One of the key ingredients is augmenting the observed data $\{(Y_i,X_i)\}_{i=1}^n$ with uniformly sampled values $\tau_i$ for $1\leq i\leq n$ which serve as training data for quantile function estimation. Using the fact that the location parameter $\mu$ in a $\tau$-tilted asymmetric Laplace distribution corresponds to the $\tau^{th}$ quantile, we build a check-loss likelihood targeting $\mu$ as the parameter of interest. We equip the check-loss likelihood parametrized by $\mu=f(X,\tau)$ with a BART prior on $f(\cdot)$, allowing the conditional quantile function to vary both in $X$ and $\tau$. The posterior distribution over $\mu(\tau,X)$ can be then distilled for estimation of the {\em entire quantile function} as well as for assessing uncertainty through the variation of posterior draws. Simulation-based predictive inference is immediately available through inverse transform sampling using the learned quantile function. The sum-of-trees structure over the conditional quantile function enables flexible distribution-free regression with theoretical guarantees. As a byproduct, we investigate posterior mean quantile estimator as an alternative to the routine sample (posterior mode) quantile estimator. We demonstrate the power of IQ-BART on time series forecasting datasets where IQ-BART can capture multimodality in predictive distributions that might be otherwise missed using traditional parametric approaches. Show Chinese abstract

Abstract: 隐式分位数 BART (IQ-BART) 在条件分位数函数上构建了一个非参数贝叶斯模型，该模型充当给定 $X$ 的 $Y$ 条件模型的模型。 其关键要素之一是使用 $1\leq i\leq n$ 的均匀采样值 $\tau_i$ 来扩充观测数据 $\{(Y_i,X_i)\}_{i=1}^n$，并将其作为分位数函数估计的训练数据。 利用位置参数$\mu$在一个$\tau$倾斜的不对称拉普拉斯分布中对应于$\tau^{th}$分位数的事实，我们构建了一个检查损失似然函数，以$\mu$作为感兴趣的参数。 我们将由$\mu=f(X,\tau)$参数化的检查损失似然与$f(\cdot)$上的 BART 先验相结合，使条件分位数函数在$X$和$\tau$上变化。 然后可以将$\mu(\tau,X)$上的后验分布提炼出来，用于估计{\em 分位数函数}以及通过后验抽样的变化来评估不确定性。 通过使用学习到的分位数函数进行逆变换抽样，可以立即获得基于模拟的预测推断。 条件分位数函数上的树结构使得可以进行灵活的无分布回归，并具有理论保证。 作为副产品，我们研究了后验均值分位数估计器，作为常规样本（后验众数）分位数估计器的替代方法。 我们在时间序列预测数据集上展示了 IQ-BART 的强大功能，其中 IQ-BART 可以捕捉预测分布中的多模态，这可能通过传统参数方法被遗漏。