The objective is two-fold. First, I introduce a 2-parameter generalization of the discrete geometric and zeta distributions. Indeed, a combination of both. It allows you to simultaneously match the variance and mean in observed data, thanks to the two parameters *p* and *α*. To the contrary, each distribution taken separately only has one parameter, and can not achieve this goal. The zeta-geometric distribution offers more flexibility, especially when dealing with unusual tails in your data. I illustrate the concept when synthesizing real-life tabular data with parametric copulas, for one of the features in the dataset: the number of children per policyholder.

Then, I show how to significantly improve grid search, and make it a viable alternative to gradient methods to estimate the two parameters *p* and *α*. The cost function — that is, the error to minimize — is the combined distance between the mean and variance computed on the real data, and the mean and variance of the target zeta-geometric distribution. Thus the mean and variance are used as proxy estimators for *p* and *α*. This technique is known as minimum contrast estimation, or moment-based estimation in statistical circles. The “smart” grid search consists of narrowing down on smaller and smaller regions of the parameter space over successive iterations.

The zeta-geometric distribution is just one example of an hybrid distribution. I explain how to design such hybrid models in general, using a very simple technique. They are useful to combine multiple distributions into a single one, leading to model generalizations with an increased number of parameters. The goal is to design distributions that are a good fit when some in-between solutions are needed to better represent the reality.

## Download the Article

The technical article, entitled *Smart Grid Search for Faster Hyperpameter Tuning*, is accessible in the “Free Books and Articles” section, here. It contains links to my GitHub files, to easily copy and paste the code. The text highlighted in orange in this PDF document are keywords that will be incorporated in the index, when I aggregate all my related articles into books about machine learning, visualization and Python. The text highlighted in blue corresponds to external clickable links, mostly references. And red is used for internal links, pointing to a section, bibliography entry, equation, and so on.

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## About the Author

Vincent Granville is a pioneering data scientist and machine learning expert, co-founder of Data Science Central (acquired by TechTarget in 2020), founder of MLTechniques.com, former VC-funded executive, author and patent owner. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. Vincent is also a former post-doc at Cambridge University, and the National Institute of Statistical Sciences (NISS).

Vincent published in *Journal of Number Theory*, *Journal of the Royal Statistical Society* (Series B), and *IEEE Transactions on Pattern Analysis and Machine Intelligence*. He is also the author of multiple books, including “Intuitive Machine Learning and Explainable AI”, available here. He lives in Washington state, and enjoys doing research on spatial stochastic processes, chaotic dynamical systems, experimental math and probabilistic number theory.

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