How would you rank the utility of Statistical Inference vs Stochastic Processes

Question

I know they're different subjects on a certain level, but would you say one begets the other? Is there a lot of information shared between the two?

Answer 1

Basically, statistical inference and (parametric) stochastic processes can be viewed as two sides of the the same coin. Let us assume that you have the following data points $\{(y_i, x_i)\}_{i=1}^n$, and you are assuming that the original stochastic process that generated these data is $Y_i=\beta_0 + \beta_1X_i + \epsilon_i$, where $X$ can be random itself or not, and $\epsilon$ follows a parametric distribution. Thus, using the tools of statistical inference you can $(1)$ estimate these ($\beta$ and $\epsilon$'s distribution) parameters, and $(2)$ then check whether your estimating method gives good results. Further, you can use numerical methods to simulate (generate) more data points.

Another example is Queuing Theory models where the parameters are unknown, here you can also use tools from statistical inference to estimate various parameter if the process.

From a didactic point of view, in order to master statistical inference on a graduate level, I think that it is not necessary to know any applied stochastic models. However, the basic prerequisites to deal with stochastic processes have a lot in common with the prerequisites for statistical inference. Namely, probability theory and some measure theoretic tools (or real analysis) are necessary for both fields. For instance, a very statistical notion like "sufficient statistic" has a definition that based purely on measure theoretic ground [1].

[1] Halmos, Paul R., and Leonard Jimmie Savage. "Application of the Radon-Nikodym theorem to the theory of sufficient statistics." The Annals of Mathematical Statistics 20.2 (1949): 225-241.