diff --git a/README.Rmd b/README.Rmd index 1c98d18..bcbf84e 100644 --- a/README.Rmd +++ b/README.Rmd @@ -23,16 +23,16 @@ knitr::opts_chunk$set( The goal of `WpProj` is to perform Wasserstein projections from the predictive distributions of any model into the space of predictive distributions of linear models. We utilize L1 penalties to also reduce the complexity of the model space. This package employs the methods as described in [Eric Dunipace and Lorenzo Trippa (2020).](https://arxiv.org/abs/2012.09999) . The Wasserstein distance is a measure of distance between two probability distributions. It is defined as: -$$ W_p(\mu,\nu) = \left(\inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^d \times \mathbb{R}^d} \|x-y\|^p d\pi(x,y)\right)^{1/p} $$ +$$W_p(\mu,\nu) = \left(\inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^d \times \mathbb{R}^d} \|x-y\|^p d\pi(x,y)\right)^{1/p}$$ where $\Pi(\mu,\nu)$ is the set of all joint distributions with marginals $\mu$ and $\nu$. In the our package, if $\mu$ is the original prediction from the original model, such as from a Bayesian linear regression or a neural network, then we seek to find a new prediction $\nu$ that minimizes the Wasserstein distance between the two: -$$ \text{argmin}_{\nu} W_{p}(\mu,\nu)^{p}, $$ +$$\mathop{\text{argmin}} _ {\nu} W _ {p}(\mu,\nu) ^ {p},$$ subject to the constraint that $\nu$ is a linear model. To reduce the complexity of the number of parameters, we add an L1 penalty to the coefficients of the linear model to reduce the complexity of the model space: -$$ \text{argmin}_{\nu} W_{p}(\mu,\nu)^{p} + P_{\lambda}(\nu), $$ +$$\mathop{\text{argmin}} _ {\nu} W _ {p}(\mu,\nu) ^ {p} + P_{\lambda}(\nu),$$ where $P_\lambda(\nu)$ is the L1 penalty on the coefficients of the linear model. diff --git a/README.md b/README.md index 46e16aa..3cb3c2b 100644 --- a/README.md +++ b/README.md @@ -15,7 +15,7 @@ described in [Eric Dunipace and Lorenzo Trippa The Wasserstein distance is a measure of distance between two probability distributions. It is defined as: -$$ W_p(\mu,\nu) = \left(\inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^d \times \mathbb{R}^d} \|x-y\|^p d\pi(x,y)\right)^{1/p} $$ +$$W_p(\mu,\nu) = \left(\inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^d \times \mathbb{R}^d} \|x-y\|^p d\pi(x,y)\right)^{1/p}$$ where $\Pi(\mu,\nu)$ is the set of all joint distributions with marginals $\mu$ and $\nu$. @@ -23,13 +23,13 @@ In the our package, if $\mu$ is the original prediction from the original model, such as from a Bayesian linear regression or a neural network, then we seek to find a new prediction $\nu$ that minimizes the Wasserstein distance between the two: -$$ \text{argmin}_{\nu} W_{p}(\mu,\nu)^{p}, $$ +$$\mathop{\text{argmin}} _ {\nu} W _ {p}(\mu,\nu) ^ {p},$$ subject to the constraint that $\nu$ is a linear model. To reduce the complexity of the number of parameters, we add an L1 penalty to the coefficients of the linear model to reduce the complexity of the model space: -$$ \text{argmin}_{\nu} W_{p}(\mu,\nu)^{p} + P_{\lambda}(\nu), $$ +$$\mathop{\text{argmin}} _ {\nu} W _ {p}(\mu,\nu) ^ {p} + P_{\lambda}(\nu),$$ where $P_\lambda(\nu)$ is the L1 penalty on the coefficients of the linear model.