R^3: I don’t see where Geisser or others making similar arguments (there have been many) address the key distinction between the causal (potential-outcome) function f(y_x;z,u) and the purely predictive (regression) function E(y;x,z,u). That is the core criticism, as I see it. The Bayesian framework does not include this key component so it has to be added on; this gap in the framework may explain why some Bayesians failed to understand the role of randomization. Interestingly, apparently in e-mails (with Pearl) toward the end of his life, Lindley recognized and conceded the need for causal extension.
In contrast, starting in the 1920s frequentists developed the necessary language for that component as it arose naturally from randomization theory, so it is rather startling how it failed to take firm hold until recent decades. Yes, experienced, intuitively smart statisticians got by without causal formalisms, and “causal inference” can be framed as a prediction problem, enabling the vast toolboxes of statistical prediction to be applied (whether frequentist, Bayesian, hybrids, etc.), e.g., see ,
Greenland, S. (2012). Causal inference as a prediction problem: Assumptions, identification, and evidence synthesis. Ch. 5 in: Berzuini, C., Dawid, A.P., and Bernardinelli, L. (eds.). Causality: Statistical Perspectives and Applications. John Wiley and Sons, Chichester, UK, 43-58.
But methodologies still need to make the fundamental distinction encapsuled as “correlation is not causation” in order to derive sound algorithms for making decisions.
I argue further that any sound statistical algorithm needs an explicit causal foundation, even if it is only a survey method, because all studies need to consider causes of observation selection and missing entries:
Greenland, S. (2022). The causal foundations of applied probability and statistics. Ch. 31 in: Dechter, R., Halpern, J., and Geffner, H., eds. Probabilistic and Causal Inference: The Works of Judea Pearl. ACM Books, no. 36, 605-624, Probabilistic and Causal Inference:The Works of Judea Pearl | ACM Books, corrected version at [2011.02677] The causal foundations of applied probability and statistics
I think recognition of this need for explicit causal models is one way the AI/computer-science literature pulled ahead of the statistics literature in the 1990s-2000s.