Black Box Variational Inference

Variational inference has become a widely used method to approximate posteriors in complex latent variables models. However, deriving a variational inference algorithm generally requires significant model-specific analysis, and these efforts can hinder and deter us from quickly developing and exploring a variety of models for a problem at hand. In this paper, we present a "black box" variational inference algorithm, one that can be quickly applied to many models with little additional derivation. Our method is based on a stochastic optimization of the variational objective where the noisy gradient is computed from Monte Carlo samples from the variational distribution. We develop a number of methods to reduce the variance of the gradient, always maintaining the criterion that we want to avoid difficult model-based derivations. We evaluate our method against the corresponding black box sampling based methods. We find that our method reaches better predictive likelihoods much faster than sampling methods. Finally, we demonstrate that Black Box Variational Inference lets us easily explore a wide space of models by quickly constructing and evaluating several models of longitudinal healthcare data

Variational approach for resolving the flow of generalized Newtonian fluids in circular pipes and plane slits

In this paper, we use a generic and general variational method to obtain solutions to the flow of generalized Newtonian fluids through circular pipes and plane slits. The new method is not based on the use of the Euler-Lagrange variational principle and hence it is totally independent of our previous approach which is based on this principle. Instead, the method applies a very generic and general optimization approach which can be justified by the Dirichlet principle although this is not the only possible theoretical justification. The results that were obtained from the new method using nine types of fluid are in total agreement, within certain restrictions, with the results obtained from the traditional methods of fluid mechanics as well as the results obtained from the previous variational approach. In addition to being a useful method in its own for resolving the flow field in circular pipes and plane slits, the new variational method lends more support to the old variational method as well as for the use of variational principles in general to resolve the flow of generalized Newtonian fluids and obtain all the quantities of the flow field which include shear stress, local viscosity, rate of strain, speed profile and volumetric flow rate. The theoretical basis of the new variational method, which rests on the use of the Dirichlet principle, also provides theoretical support to the former variational method.Comment: 22 pages, 6 figures, 5 table

A variation equation for the wave forcing of floating thin plates

A variational equation is derived for a floating thin plate subject to wave forcing. This variational equation is derived from the thin plate equations of motion by including the forcing due to the wave through the integral equation derived using the free surface Green’s function. This equation combines the optimum method forsolving the motion of a thin plate (the variational equation) with the optimum method for solving the wave forcing of a floating body (the Green’s function method). Solutions of the variational equation are presented for some simple thin plate geometries using polynomial basis functions. The variational equation is extended to the case of plates of variable properties and to multiple plates and example solutions are presented

A field theoretic approach to master equations and a variational method beyond the Poisson ansatz

We develop a variational scheme in a field theoretic approach to a stochastic process. While various stochastic processes can be expressed using master equations, in general it is difficult to solve the master equations exactly, and it is also hard to solve the master equations numerically because of the curse of dimensionality. The field theoretic approach has been used in order to study such complicated master equations, and the variational scheme achieves tremendous reduction in the dimensionality of master equations. For the variational method, only the Poisson ansatz has been used, in which one restricts the variational function to a Poisson distribution. Hence, one has dealt with only restricted fluctuation effects. We develop the variational method further, which enables us to treat an arbitrary variational function. It is shown that the variational scheme developed gives a quantitatively good approximation for master equations which describe a stochastic gene regulatory network.Comment: 13 pages, 2 figure