How Bayesian Inference Tackles Made-Up, Magical, or Supernatural Hypotheses

Introduction to Bayesian Inference

In the field of statistical inference, Bayesian inference provides a principled approach to updating beliefs based on observed evidence. When dealing with complex hypotheses, such as those involving magical or supernatural phenomena, the framework of Bayesian inference offers a rigorous way to navigate and evaluate these claims. This article delves into how Bayesian inference handles these hypotheses by examining the role of prior probabilities and model comparison.

Bayesian Inference and Model Comparison

Bayesian inference is not confined to a single model. It requires comparing several plausible models to understand which one is most likely to explain the observed data. This is done by using Bayes' rule to compute the posterior probabilities of each model. The key equation for this process is:

Bayes' Rule: PMi|D ∝ PD|Mi PMi

This equation shows that the posterior probability of a model is proportional to the likelihood of the model given the data multiplied by the prior probability of the model.

Handling Supernatural Hypotheses in Bayesian Inference

Suppose you encounter a scenario where a supernatural hypothesis is proposed, such as 'The observations are the way they are because they had to be that way.' This implies that the probability of the data given the model is 1, as no other observation would be possible under this model. However, this single model does not provide a realistic framework for comparison due to its complete unwavering nature.

For instance, consider the example of a sequence of 15 coin flips: HTTHTHTHHTHTTHT. Several probabilistic models might be proposed to explain this data, such as independent Bernoulli trials, unknown p inferred from data, or a Markov chain with memory effects. If no data had been observed, these models would all be roughly equally plausible. However, a single, deterministic supernatural explanation would assign a prior probability of 1/N, where N is the number of possible sequences, making it highly unlikely.

Critique of Unfavorable Prior Probabilities

The argument raises a critical point about the prior probabilities assigned to different models. It is argued that supernatural hypotheses are too rigid and deterministic, and thus, they should be assigned lower prior probabilities. Consider the scenario of the coin flips sequence. The rationale would be that if we haven’t seen the data, a particular sequence (say HTTHTHTHHTHTTHT) is just as likely as any other sequence. Assigning it a higher prior probability would be illogical.

Philosophical Considerations and Bayesian Determinism

From a philosophical standpoint, some might argue for physical determinism, where every event is the causal result of preceding events. In this view, the set of all possible sequences could indeed be considered models. Bayesian inference would then assign equal prior probabilities to all these models, and the model that happens to match the observed sequence would have the highest posterior probability.

This argument, however, might strike some as strange because it relies on an assumption of complete determinism, which is not universally accepted. Moreover, it highlights the importance of prior beliefs in Bayesian inference. If these prior beliefs are too rigid and unyielding, Bayesian inference will reflect them, leading to a confirmation bias. In essence, 'Garbage in, Garbage out.'

Conclusion

Bayesian inference provides a robust framework for evaluating complex hypotheses, including those considered magical or supernatural. By emphasizing the role of prior probabilities and model comparison, it helps avoid over-reliance on singular, deterministic explanations. While Bayesian inference can handle a wide range of models, it is crucial to choose sensible prior probabilities that reflect our understanding of the underlying phenomena. This approach ensures that the conclusions drawn are both robust and reflective of the available data.