# What is the underlying standard Normal space?

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Uncertain model parameters can be represented by a large variety of probability distributions. This includes for example the Normal, log-Normal, Gumbel, Weibull, beta or uniform distribution. Moreover, some uncertain model parameters might be statistically dependent. The parameters can also differ considerably with respect to their mean values or standard deviations.

If numerical methods are applied to probabilistic models, it is sometimes preferable to transform the model parameters to a standardized probability space. A popular strategy is to transform the uncertain model parameters $\mathbf{X}$ into statistically independent standard Normal random variables $\mathbf{U}$; i.e., $T:\mathbf{X}\to\mathbf{U}$. The space of the parameters $\mathbf{X}$ is then referred to as the physical/original space and the space associated with $\mathbf{U}$ is referred to as the underlying standard Normal space. The inverse transformation $T^{-1}:\mathbf{U}\to\mathbf{X}$ is used to generate a sample in $\mathbf{X}$-space from a realization of independent standard Normal random variables. Sampling methods based on this principle are referred to as transformation methods, this includes the Nataf transformation and the Rosenblatt transformation.

Let's denote the model by $h(\mathbf{X})$. The numerical method applied to $h(\cdot)$ actually works in $\mathbf{U}$-space and sees the model as $h(T^{-1}(\mathbf{U})) = H(\mathbf{U})$. Thus, from the perspective of the numerical method, all model parameters are expressed as standard Normal random variables, and the transformation $T^{-1}$ is actually part of the model $H$.

The advantage of working in the underlying standard Normal space is that numerical algorithms can be designed independently of the probabilistic model employed. This allows us to set up importance sampling densities or Markov chain proposal distributions that achieve acceptable performance for a wide range of problems. In this context, it is also helpful that the domain of $\mathbf{U}$ is unbounded, whereas the domain of $\mathbf{X}$ can be bounded (depending on the type of the probability distribution of $\mathbf{X}$).

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