A family of pdfs or pmfs is called an exponential family if $f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$
What is the motivation of this definition? It seems that if would be easier to verify properties of pdfs such as sufficiency if they are of this form? So did people come up with the definition of exponential family after Neyman and Pearson came up with the factorization theorem?