Bayes, buttressed by design-based ideas, is the best overarching paradigm for sample survey inference
Section 2. Notation, and a seminal paper

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In this section, I introduce some notation and a seminal paper that underlies much of the thinking in this paper. Let $Y=\left(y_1,\dots ,y_N\right)$ and $S=\left(S_1,\dots ,S_N\right)$ where $N<\infty$ is the number of units in the population, $y_i$ is the set of survey variables and $S_i$ is the selection indicator for the $i^{\text{th}}$ unit, with value 1 when the $i^{\text{th}}$ unit is selected and 0 otherwise. Let $Z$ represent design information such as stratum or cluster indicators, and $z_i$ the value of $Z$ for unit $i.$ Consider inference about a finite population quantity $Q\left(Y,Z\right),$ for example the population total $Q\left(Y,Z\right)={\displaystyle {\sum }_{i=1}^N{y}_i},$ where $Y=\left(y_1,\dots ,y_N\right).$ A general model-based approach treats both $S$ and $Y$ as random variables, with joint distribution given $Z:$

$$f_{S,Y|Z}\left(S,Y|Z,\theta ,\psi \right)=f_{Y|Z}\left(Y|Z,\theta \right)f_{S|Y,Z}\left(S|Z,Y,\psi \right),(2.1)$$

where $f_{Y|Z}$ represents the density of survey variables $Y$ indexed by unknown parameters $\theta ,$ and $f_{S|Y,Z}$ represents the model for inclusion indexed by unknown parameters $\psi .$ For a probability sample with no nonresponse, the sampling distribution is known and does not depend on $Y,$ that is,

$$f_{S|Y,Z}\left(S|Z,Y,\psi \right)=f_{S|Z}\left(S|Z\right);(2.2)$$

design-based methods base inferences on the distribution of statistics in repeated sampling from this distribution.

For a survey with unit nonresponse, inclusion occurs when a unit is selected, and then responds given selection. Accordingly, let $R_i=1$ if selected unit $i$ responds and $R_i=0$ otherwise. The model-based approach models the joint distribution of $S,$ $R$ and $Y$ given $Z$ as

$$f_{S,R,Y|Z}(S,R,Y|Z,\theta ,\psi )=f_{Y|Z}\left(Y|Z,\theta \right)f_{S|Y,Z}\left(S|Z,Y,\psi \right)f_{R|S,Y,Z}\left(R|Z,Y,S,\varphi \right),(2.3)$$

adding to equation (2.1) a model for unit nonresponse with density $f_{R|S,Y,Z}.$ Item nonresponse can also be treated by modeling indicators for the patterns of item missingness (e.g., Little, 2003b).

Treating $S,$ $R$ and $Y$ as random variables is a key feature of Rubin (1978), which I regard as one of the landmark statistics papers in the history of statistics. The paper provides conditions under which the missingness and selection mechanisms are ignorable, that is, do not need to be modeled for likelihood-based inference, extending definitions of ignorability for missing data in Rubin (1976), while providing a framework for inference when selection and/or missingness is non-ignorable. The significance of the paper for survey sampling is easily missed, because its main focus is on the role of the treatment assignment mechanism in the context of inference about causal effects. The assignment mechanism is ignorable under random treatment assignment, as in randomized clinical trials. The paper thus lays a general framework for causal inferences comparing treatments, and it is for this feature that the paper is best known. However, the paper also provides a Bayesian justification for random sampling, as a means of avoiding the need for a model for selection.

In frequentist superpopulation modeling (e.g., Valliant, Dorfman and Royall, 2000), the parameters in models are treated as fixed; in Bayesian survey modeling, these parameters are assigned a prior distribution, and inferences for $Q\left(Y\right)$ are based on its posterior predictive distribution given the data. In large samples, the prior distribution plays a minor role, and the two approaches yield similar answers for comparable models; in particular the ML estimate of a parameter is essentially the mode of the posterior distribution under a uniform prior, and as such has a Bayesian interpretation. In small samples, uncertainty about the model parameters is propagated when they are integrated out of the posterior distribution. This approach to propagating error in parameters allows Bayesian inferences for judiciously chosen models and priors to be better calibrated than inferences from superpopulation modeling inferences, in a sense of having better frequentist properties in repeated sampling (Rubin, 1978). So, in my opinion “superpopulation modeling is super, but Bayes is better”.

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