fix a typo

paper/paper.md

@@ -51,7 +51,7 @@ The primary goal in GPCERF is to find appropriate values for the hyper-parameter
 
 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. 
 
-The scaling parameter $\alpha$ and $\beta$ determine how much information the estimation will draw from the two coordinates: GPS score ($s(W, X)$) and exposure level ($W$). A large scaling parameter suggests that varying the corresponding coordinates is only associated with a minor change in the outcome, that is, this coordinate does not contribute too much to the variation of the outcome. The noise-to-signal parameter $\gamma$ encodes how different observed data is from pure noise.  A large $\gamma$ indicates strong associations between the outcome and the coordinates of GP while a small $\gamma$ suggests the observed outcome is likely to be drawn from a random process that is independent of the coordinates. In the setting of observational studies and under the no unobserved confounding assumption, which GPCERF is specifically designed for, both the exposure level and the GPS score encode important information for the estimation of CERF. As a result, the range of their scaling parameter should be comparable and the covariate balance will determine which coordinate is more important (smaller scaling factor). The range should also cover both ends of the importance from extremely important to nearly irrelevant. We choose to achieve this by considering the range on the $log_{10}$ scale with equally spaced candidate values. The range of  also follows the same strategy when the prior belief about the strength of causal effect of the exposure is weak.
+The scaling parameter $\alpha$ and $\beta$ determine how much information the estimation will draw from the two coordinates: GPS score ($s(W, X)$) and exposure level ($W$). A large scaling parameter suggests that varying the corresponding coordinates is only associated with a minor change in the outcome, that is, this coordinate does not contribute too much to the variation of the outcome. The signal-to-noise ratio parameter $\gamma/\sigma$ encodes how different observed data is from pure noise.  A large $\gamma/\sigma$ indicates strong associations between the outcome and the coordinates of GP while a small $\gamma/\sigma$ suggests the observed outcome is likely to be drawn from a random process that is independent of the coordinates. In the setting of observational studies and under the no unobserved confounding assumption, which GPCERF is specifically designed for, both the exposure level and the GPS score encode important information for the estimation of CERF. As a result, the range of their scaling parameter should be comparable and the covariate balance will determine which coordinate is more important (smaller scaling factor). The range should also cover both ends of the importance from extremely important to nearly irrelevant. We choose to achieve this by considering the range on the $log_{10}$ scale with equally spaced candidate values. The range of  also follows the same strategy when the prior belief about the strength of causal effect of the exposure is weak.
 
 In the estimation step, the optimal parameters are used to estimate the posterior mean and standard deviation of $R(w)$ at a set of exposure values of interest. The outcome data is not used during the tuning step, separating the design and analysis phases. @Ren_2021_bayesian discusses the  implemented approaches in detail. In the following we provide an example for running the package for each implemented models.