Bayes, buttressed by design-based ideas, is the best overarching paradigm for sample survey inference
Section 1. Introduction

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Bayesian inference is in my view the best overarching inferential paradigm for statistical inference from surveys, whether from probability or non-probability samples. See for example Ericson (1969), Binder (1982), Rubin (1987), Ghosh and Meeden (1997), Little (2003ab, 2004, 2012, 2015), Sedransk (2008) and Fienberg (2011). However, design-based properties of Bayesian inferences are important, because “all models are wrong”, and broad acceptance of results requires inferences that have good operating characteristics in repeated sampling. In particular, Bayesian models need to incorporate complex design features to yield inferences that are approximately calibrated, in the sense that credible intervals have close to nominal levels when treated as confidence intervals in repeated sampling (Rubin, 1984, 2019; Little, 2006). In large samples, flexible working models can avoid strong parametric assumptions that lead to potentially biased estimates.

To focus discussion, consider the problem of deriving a point estimate $q$ of a finite population quantity $Q,$ and a 95% interval estimate $I_{0.95}=\left(l,u\right)$ that captures uncertainty in $q,$ the interval may have a frequentist interpretation as a 95% confidence interval, or a Bayesian interpretation as a 95% posterior credible interval for $Q.$ I think scientists who are not statisticians generally interpret the interval $I$ in a Bayesian way, as a fixed interval capturing the uncertainty about $Q.$ However, I do not focus unduly on the difference in interpretation of $I_{0.95}$ under the two paradigms. The 95% nominal value is by convention and other levels could be chosen.

An appealing feature of finite population survey sampling is that it deals with real (though unknown) quantities. For “analytic” survey inference, where the focus is on parameters of idealized models of the population, such as regression coefficients in a multiple regression model, define the finite population quantity $Q$ as the estimate of the parameter of interest if the model was fitted to data for the whole population, according to some agreed fitting method such as least squares or maximum likelihood (ML). A useful feature of this construction is that $Q$ is then a real quantity, rather than a feature of a simplified hypothetical model of the population (e.g., Little, 2004).

Under the Bayesian paradigm, inference for $Q$ is based on its posterior predictive distribution given the data, for judicious choices of model and prior distribution for unknown parameters. Thus $q$ may be the posterior predictive mean of $Q,$ and $I_{0.95}$ the 2.5^th to 97.5 percentile of the posterior predictive distribution, or the limits of the range of values of $Q$ with the highest posterior density, assuming the posterior predictive distribution is unimodal. A useful feature of the Bayesian approach is that “finite population corrections” are automatically incorporated in the posterior predictive distribution of finite population quantities $–$ as the sample converges to the finite population, the posterior variance tends to zero.

The focus is on developing suitable models and prior distributions. Computation used to be a major challenge and is still a practical consideration, though less so now with the advent of Markov Chain Monte Carlo methods and rapid advances in Bayesian computation. Thus, the complaint that Bayes is conceptually appealing but simply too difficult to implement is harder to sustain than it was, say, thirty years ago.

The remainder of the paper is organized as follows. In Section 2, I introduce some notation and describe formally the models and prior distributions required by the Bayesian approach to survey data, with and without nonresponse. In Section 3, I describe generally desirable features of an inference about $Q,$ and discuss why I believe the Bayesian paradigm for suitably-chosen models can be more successful at achieving these features than the design-based approach. In Section 4 I present a variety of examples, intended to illustrate the points in Section 3. I conclude in Section 5 by proposing ten reasons to be Bayesian in the survey sampling setting.

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