bulkem: Background
18 Jun 2015

The inverse Gaussian distribution

The inverse Gaussian distribution is an exponential-family probability distribution with the density function:

$$ f(x)=\left[\frac{\lambda}{2\pi x^{3}}\right]^{1/2}\exp\frac{-\lambda(x-\mu)^{2}}{2\mu^{2}x} $$

for \( x>0 \), mean \( \mu>0 \) and shape \( \lambda>0 \) (Seshadri, 1993, p. 1).

The expectation-maximisation algorithm

The EM algorithm (Dempster et al., 1977) iteratively refines a maximum likelihood estimate in the presence of missing data. Here, we use it to fit mixture models, as described in Bilmes (1998). Two characteristics are of note:


A GPU is a component of a computer that accelerates graphics operations. All modern computers and mobile devices include a GPU. GPUs can be programmed to perform general-purpose computations in the same way as a CPU; this is referred to as ‘general purpose GPU computing’.

GPUs are similar to CPUs in that they run user-defined software. The key difference is that while a CPU generally has a small number of execution units (two or four are common for consumer hardware), a GPU may have thousands of execution units operating in parallel. The CPU is optimised for serial operations – performing a sequence of instructions as quickly as possible. The GPU is optimised for parallel operations where the same instructions are performed many times on different data. Each execution unit of the GPU is simple and more restricted, but there are many more of them. The peak computational output (measured in instructions per second) is far greater than for a CPU. Redesigning the problem to take advantage of this structure is the major challenge of the programmer working on a GPU computing problem.

There are two major standards for GPU computing: OpenCL and CUDA. OpenCL is supported by most GPU vendors. CUDA is only supported by NVIDIA hardware but it is a mature standard with excellent tools and documentation. We will only consider CUDA from this point on.

There are significant barriers to widespread adoption of GPU computing.


M Aitkin and GT Wilson. Mixture models, outliers, and the EM algorithm. Technometrics, 22(3): 325–331, Aug 1980.

JA Bilmes. A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models. Technical Report TR-97-021, International Computer Science Institute, 1998.

AP Dempster, NM Laird, and DB Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1):1–38, 1977.

V Seshadri. The inverse gaussian distribution: a case study in exponential families. Oxford Science Publications, 1993.

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