In a paper I've been reading recently, at some point the authors define a new probability distribution over original data $\mathcal{D} =\{\textbf{x}_1, \dots, \textbf{x}_N\}$, by introducing a gaussian perturbation.
In practice, given that we have some (unknown) probability $p_{data}(\textbf{x})$ and some scalar $\sigma >1$, the new (perturbed) probability is defined as
$$q_{\sigma}(\textbf{x}) = \int p_{data}(\textbf{t}) \mathcal{N}(\textbf{x} | \textbf{t} \, ,\sigma^2I) \, d\textbf{t} \, \, .$$
Based on this notation, what I understand is that, if my initial data point is distributed according to the original probability $p_{data}$, what we do now is considering $\tilde{\textbf{x}} = \textbf{x} \cdot \varepsilon$ where $\epsilon \sim \mathcal{N}(\textbf{t}, \sigma^2I)$ and therefore the new density function takes into account the introduced noise. Nevertheless, it's not clear what the role of $\textbf{t}$ is in this context and why we integrate over it. Additionally, how can we ensure that the new defined distribution $q_\sigma(\textbf{x})$ integrates to $1$?
