I arrived at and then developed the analogical approach to statistical inference through a path that is marked out by my published papers.
As is still true for nearly everyone at the present time who thinks about how to carry out statistical inference, all my ideas about statistical inference before 1999 lay in classical statistical theory (i.e. empirical theory, axiomatic theory or a hybrid of these two theories). All that I can say in my defence is that I was much more dissatisfied with classical statistical theory than most people that I knew.
The first sense I had that there was something very important that lay beyond classical statistical theory came via a re-evaluation of R. A. Fisher’s ideas on fiducial inference in late 1999. This led me to become a critic of axiomatic justifications for Bayesian inference (Bowater 2004). Also, it made me realise that a new form of fiducial inference could only be justified via a definition of subjective probability that relied on making analogies to standard physical experiments. Once this type of probability had been clarified (Bowater 2017a), the remaining development of this new form of fiducial inference, namely subjective fiducial inference, could then be completed (Bowater 2017b). Nevertheless, at that time, there was a severe limit on the applicability of subjective fiducial inference as it appeared that it could not be generally extended to deal with problems of inference involving more than one unknown parameter.
The breakthrough in being able to extend subjective fiducial inference to multi-parameter problems of inference came with the realisation that this could be achieved on the basis of the full conditional fiducial distributions by using an analytical approach or a Gibbs sampling method, with the latter method not requiring these conditional distributions to be compatible (Bowater 2018a). Nevertheless, despite this breakthrough, multivariate subjective fiducial inference could not handle discrete data, e.g. event counts.
After a short while, the problem in handling discrete data was solved by combining subjective fiducial inference and Bayesian inference, which effectively involved combining what were later defined as the Bayesian and fiducial analogies. The type of inference developed, namely organic fiducial inference (Bowater 2019a, Bowater 2021b), also allowed pre-data knowledge about model parameters to be taken into account, which was effectively achieved using what would later be called the artificial data analogy.
The type of subjective probability mentioned earlier that underlay subjective and later organic fiducial inference continued to be developed, simplified and clarified, culminating in it being named analogical probability under its most recent definition (Bowater 2018b, Bowater 2022a).
The issue of how to learn from observed data about the probability of a sharp hypothesis (an hypothesis that a parameter is equal or close to a given value) is a thorny problem. I mistakenly felt that this problem could be solved using P values (Bowater and Guzman 2019, Bowater 2019b, Bowater 2020, Bowater 2021). I now accept that essentially P values solve nothing (Bowater 2024). It was only a breakthrough that enabled me to solve this problem in a methodologically sound way by combining Bayesian and fiducial inference (Bowater 2022b). This breakthrough also made me appreciate the fundamental importance that analogy making has in statistical inference as the solution relies on a subtle interplay between the fiducial and Bayesian analogies that avoids compromising their integrity.
As alluded to above, multivariate subjective fiducial inference is based on constructing joint post-data distributions on the basis of already determined full conditional fiducial distributions. This idea was further developed in a number of papers (Bowater 2020, Bowater 2021a, Bowater 2023) to allow these full conditional distributions to be any type of post-data distributions constructed using a justifiable method of inference. In the most recent formulation of this idea, called the fiducial-Bayesian fusion (Bowater 2023), each of the full conditional post-data distributions can be constructed using organic fiducial inference only, Bayesian inference only or a combination of these two methods of inference. Nevertheless, I later was able to appreciate that it is not organic fiducial inference and Bayesian inference that are of fundamental importance in statistical inference, but the quality of the statistical analogies on which any method of inference is based.
It took me a while to realise how fundamental analogy making was in statistical inference, but when this realisation finally came so many things fell into place that I feel entitled to refer to this type of inference as modern statistical theory (Bowater 2025). I have identified key statistical analogies, namely the pre-post event analogy, the fiducial analogy, the general sampling analogy, the Bayesian analogy, the artificial data analogy and the statistical modelling analogy that together justify a general theory of statistical inference (Bowater 2025). Also, by using the framework of analogy-based inference, I have demonstrated the flaws and weaknesses of statistical methods that have received much attention, e.g. posterior, partial, fractional and intrinsic Bayes factors, P values and Fisher’s fiducial inference in its final form (Bowater 2025). Further developments of this framework will continue to be made, and hopefully, for the benefit of the discipline of statistics, it will not just be me who will be making them.
Note: To view any particular paper, click on its arXiv number or page range. Colour code: Black – I still endorse; Blue – I still partially endorse; Red – I no longer endorse.
Bowater, R. J. (2025). Analogy making as the basis of statistical inference. arXiv.org (Cornell University), Statistics, arXiv:2504.16186. Published: 22-Apr-2025.
Bowater, R. J. (2024). Probabilistic inference when the population space is open. arXiv.org (Cornell University), Statistics, arXiv:2410.12930. Published: 16-Oct-2024.
Bowater, R. J. (2023). The fiducial-Bayes fusion: A general theory of statistical inference. arXiv.org (Cornell University), Statistics, arXiv:2310.01533. Published: 02-Oct-2023.
Bowater, R. J. (2022b). Sharp hypotheses and organic fiducial inference. arXiv.org (Cornell University), Statistics, arXiv:2207.08882. First version: 18-July-2022. Current version: 14-Sep-2023.
Bowater, R. J. (2022a). Physical, subjective and analogical probability. arXiv.org (Cornell University), Statistics, arXiv:2204.10159. Published: 20-Apr-2022.
Bowater, R. J. (2021b). A revision to the theory of organic fiducial inference. arXiv.org (Cornell University), Statistics, arXiv:2111.09279. Published: 17-Nov-2021.
Bowater, R. J. (2021a). A very short guide to IOI: a general framework for statistical inference summarised. arXiv.org (Cornell University), Statistics, arXiv:2104.11766. Published: 23-Apr-2021.
Bowater, R. J. (2020). Integrated organic inference (IOI): a reconciliation of statistical paradigms. arXiv.org (Cornell University), Statistics, arXiv:2002.07966. First version: 19-Feb-2020. Current version: 15-Apr-2021 (Final version with corrections).
Bowater, R. J. (2019). Sharp hypotheses and bispatial inference. arXiv.org (Cornell University), Statistics, arXiv:1911.09049. First version: 20-Nov-2019. Current version: 25-Jan-2021 (Final version).
Bowater, R. J. and Guzmán-Pantoja, L. E. (2019). Bayesian, classical and hybrid methods of inference when one parameter value is special. Journal of Applied Statistics, 46, 1417-1437. The authors' version of this paper can be found here.
Bowater, R. J. (2019). Organic fiducial inference. arXiv.org (Cornell University), Statistics, arXiv:1901.08589. First version: 23-Jan-2019. Current version: 08-Apr-2021 (Final version with corrections).
Bowater, R. J. (2018b). On a generalised form of subjective probability. arXiv.org (Cornell University), Statistics, arXiv:1810.10972. First version: 25-Oct-2018. Current version: 24-Mar-2022 (Final version with minor corrections).
Bowater, R. J. (2018a). Multivariate subjective fiducial inference. arXiv.org (Cornell University), Statistics, arXiv:1804.09804. First version: 25-Apr-2018. Current version: 07-Apr-2021 (Final version with corrections).
Bowater, R. J. (2017b). A defence of subjective fiducial inference. AStA Advances in Statistical Analysis, 101, 177-197. The author's version of this paper can be found here.
Bowater, R. J. (2017a). A formulation of the concept of probability based on the use of experimental devices. Communications in Statistics: Theory and Methods, 46, 4774-4790. The author's version of this paper can be found here.
Bowater, R. J. (2004). A foundational justification for a weighted likelihood approach to inference (with discussion). International Statistical Review, 72, 307-336.
(Note: I never really thought weighted likelihood inference was a good idea, but I used this paper as a vehicle to discuss the meaning of probability and to attack some important principles that underlie Bayesian theory.)