By Dunson D.B., Herring A.H.

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**Extra resources for Bayesian latent variable models for mixed discrete outcomes**

**Sample text**

Cushman (1987) made similar conclusions after an in-depth analysis of the perturbation solutions provided by Gelhar and Axness (1983). The assumption that hydraulic conductivity is the only source of uncertainty in many existing solutions has to be challenged. Dispersion in natural formations can not only be explained by the variability of hydraulic conductivity; and if this is feasible the large-scale hydraulic conductivity can safely be described by well defined deterministic functions. Some of the solutions are only valid in the steady state of the concentration field.

49) as the definition of Ito integral for the purpose of this book. As stated earlier I[X](w) is a stochastic process and it has the following properties (see, for example, 0ksendal (1998) for more details): I. [to integral is linear. If X(t) and Y(t) are predictable processes and a and then 2. P as some constants, Ito integral has zero-mean property E(I[X](w))= 0 . 3. Ito integral is isometric. The isometry property says that the expected value of the square of Ito integral is the integral with respect to t of the expectation of the square of the process X (t).

In this way we will show that Fickian assumptions may be avoided and a more fundamental description addressing, for example, the scale dependence problem can be attempted. We are incorporating Brownian motion, a mathematical object or process representing variability in the system and the conservation laws in the stochastic dynamical description of the problem. Before this idea can be implemented, it is first necessary to outline aspects of the theory of stochastic differential equations and some essential elements of stochastic processes that will be used.

### Bayesian latent variable models for mixed discrete outcomes by Dunson D.B., Herring A.H.

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