The concept of parameter space is found in the theory of statistical inference. In a statistical inference problem, the statistician utilizes a sample to understand from what probability distribution the sample itself has been generated. Attention is usually restricted to a well-defined set of probability distributions that could have generated the sample. When these probability distributions are put into correspondence with a set of real numbers (or real vectors) such set is called the parameter space and its elements are called parameters.
A detailed presentation of the concepts of parameter and parameter space can be found in the lecture entitled Statistical inference.
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