Abstract
Bayesian inference , otherwise known as “probability theory”, is the theory of how to combine uncertain information from multiple sources to make optimal decisions under uncertainty. These sources include empirical data and also other beliefs, called priors, which may come from previous experiments, theory, and subjective estimates. Bayesian theory makes probabilistic inferences which are complete probability distributions over our beliefs about unobserved variables of interest, including generative parameters of models assumed to cause to data, as well as unobserved variables which are caused by the observed data. Bayesian inference is computationally hard, so typically works with approximate calculations on large compute systems. Bayesian inference is provably (Bernardo and Smith 2001) the only system able combine beliefs to make decisions consistently and optimally.
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Notes
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Bayes’ theorem is also almost certainty misnamed in its discoverer too. The leading history of this reads like a detective novel itself (Stigler, Stephen M. Who discovered Bayes’ theorem?, The American Statistician 37.4a (1983):290–296) and concludes that the discovery is most probably due to Nicholas Saunderson , who grew up in a village 25 miles from ITS Leeds, and – amazingly – performed all of his mathematics research after going blind, using a tactile abacus of his own invention.
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If this sounds far-fetched, a statistically similar miscarriage of justice actually happened in 1999, sentencing Sally Clerk to two life sentences for what was later proven in (R. vs. Clerk. 2003), using Bayesian analysis, to be two coincidental cot deaths.
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The other main standard approximations are “loopy belief propagation”, “Variational Bayes”, and assuming various distributions to make exact inference tractable.
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Academic paper publications are not completely free from these problems in practice, as their values are also measured in cash values by modern university managers. Bayesian data scientists make up a large part of the “ fake science” movement which seeks to expose scientific malpractice, such as reporting of false results or statistical interpretations for financial gain. They argue that while Science’s theoretical utility lies in reporting the truth, individual researchers are more strongly motivated by the needs to publish and advance their careers under current research management models. See www.callingbullshit.org.
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Even the collapse of quantum wave functions can be set up in this way in relativistic Quantum Field Theory, which goes beyond non-reversible Quantum Mechanics.
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Controller Area Network, a common networking standard for communication between electronic devices inside a vehicle.
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Usually you can do even better than this “Bayesian Model Averaging” using other types of model combination. Averaging is only optimal if you are sure that your hypothesis set contains the ground truth rather than just approximations (P. Domingos, Bayesian Averaging of Classifiers and the Overfitting Problem, International Conference on Machine Learning (ICML), 2000).
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Thanks to ITS Leeds student Panagiotis Spyridakos for porting this code to PyMC3.
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Fox, C. (2018). Bayesian Inference. In: Data Science for Transport. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-72953-4_6
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DOI: https://doi.org/10.1007/978-3-319-72953-4_6
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