We propose a novel and practical privacy notion called $f$-Membership Inference Privacy ($f$-MIP), which explicitly considers the capabilities of realistic adversaries under the membership inference attack threat model. Consequently, $f$-MIP offers interpretable privacy guarantees and improved utility (e.g., better classification accuracy). In particular, we derive a parametric family of $f$-MIP guarantees that we refer to as $\mu$-Gaussian Membership Inference Privacy ($\mu$-GMIP) by theoretically analyzing likelihood ratio-based membership inference attacks on stochastic gradient descent (SGD). Our analysis highlights that models trained with standard SGD already offer an elementary level of MIP. Additionally, we show how $f$-MIP can be amplified by adding noise to gradient updates. Our analysis further yields an analytical membership inference attack that offers two distinct advantages over previous approaches. First, unlike existing state-of-the-art attacks that require training hundreds of shadow models, our attack does not require any shadow model. Second, our analytical attack enables straightforward auditing of our privacy notion $f$-MIP. Finally, we quantify how various hyperparameters (e.g., batch size, number of model parameters) and specific data characteristics determine an attacker's ability to accurately infer a point's membership in the training set. We demonstrate the effectiveness of our method on models trained on vision and tabular datasets.