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paper/paper.md

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 where $h : [0, \infty) \rightarrow [0, 1]$ is a non-increasing function; and $\alpha$ and $\beta$ define the relative importance of GPS and exposure values, respectively. $\gamma$ indicates the scale of the GP. We call the collection $(h, \alpha, \beta, \gamma)$ the hyper-parameters of the GP.
 
-The primary goal in GPCERF is to find appropriate values for the hyper-parameters. In the context of causal inference,  ''appropriate'' values of the hyper-parameters are those that make the estimator of CERF as if it is generated from a study with randomized design. To be more concrete, note that the GP estimates $R(w)$ by creating a pseudo-population that is a weighted version of the original dataset [see more details in @Ren_2021_bayesian]. The weight for each sample in the original dataset is a function of the hyperparameters. By tuning the hyperparameters, we can minimize the sample correlations between $W$ and each component of $C$ in this pseudo-population, rendering the pseduo-population to be more balanced on these covariates $C$. In practice, we minimize the covariate balance, which is a summary of the sample correlations between $W$ and each of $C$ to tune our hyper-parameters. Covaraite balance is computed by assessing the correlation between $W$ and $C$ in the pseudo-population using the _wCorr_ R package [@wCorr_R].
+The primary goal in GPCERF is to find appropriate values for the hyper-parameters. In the context of causal inference,  ''appropriate'' values of the hyper-parameters are those that make the estimator of CERF as if it is generated from a study with randomized design. To be more concrete, note that the GP estimates $R(w)$ by creating a pseudo-population that is a weighted version of the original dataset [see more details in @Ren_2021_bayesian]. The weight for each sample in the original dataset is a function of the hyperparameters. By tuning the hyperparameters, we can minimize the sample correlations between $W$ and each component of $C$ in this pseudo-population, rendering the pseudo-population to be more balanced on these covariates $C$. In practice, we minimize the covariate balance, which is a summary of the sample correlations between $W$ and each of $C$ to tune our hyper-parameters. Covariate balance is computed by assessing the correlation between $W$ and $C$ in the pseudo-population using the _wCorr_ R package [@wCorr_R].
 
 Both GP and nnGP approaches involve two primary steps - tuning and estimation. GPCERF conducts a grid search on the range of provided $\alpha$, $\beta$, and $\gamma/\sigma$. The kernel function is also selected if the user provides multiple candidates. During the tuning step,  covariate balance is minimized by choosing the optimal hyperparameters.