Question

In Bayesianism, every belief in a hypothesis is updated in the same way. You have a prior probability P (H). You have the probability of an observation under a hypothesis P (E|H). And then you update your P(H) using those two factors and P (~H) and P(E|~H) where ~H = the hypothesis not being true.

Now, most Bayesians, even the most objective of ones, refuse to assign a zero prior in general. They recommend assigning a non zero prior to any logically coherent hypothesis. Their reasoning is that otherwise, no amount of evidence would change your mind.

Here is an example of a logically coherent hypothesis: Adam can guess, using psychic powers, what the price of each stock in the world will be at the end of each trading day.

As you can hopefully tell, this isn’t much of a good explanation. It doesn’t explain how he would do this. What would "psychic powers" mean here? What would be the mechanism?

Answer 1

Bayesian beliefs about hypotheses are updated based on evidence, avoiding zero probabilities to ensure that evidence can influence beliefs. 'Psychic powers' is a hypothesis that, while testable in theory, lacks a defined mechanism, yet it receives a non-zero prior in Bayesian analysis to allow for belief updating.

Step-by-step explanation:

Beyond Zero Probabilities in Bayesianism

In the Bayesian framework, beliefs about the truth of hypotheses are updated based upon evidence, where the belief in the hypothesis before observing the data is called the prior probability. Bayesians often discourage assigning a zero probability to any hypothesis that is logically coherent, as doing so would mean that no amount of evidence could cause us to update our belief in that hypothesis. Take, for example, the hypothesis that 'Adam can guess what the price of stock will be at trading day's end through psychic powers.' While this hypothesis can be articulated clearly and does not involve logical contradictions, it presents a challenge in terms of its scientific testability, as the mechanism of 'psychic powers' is not defined or understood within the existing scientific paradigm. Nevertheless, in Bayesian analysis, we would still assign a non-zero prior probability to it, allowing for the possibility that future evidence could update our belief in its likelihood.

The value of incorporating prior knowledge and evidence lies in the Bayesian paradigm's flexibility and its embrace of uncertainty as an inherent aspect of parameter estimates, which contrasts with classical hypothesis testing that seeks to disprove null hypotheses. Bayesian methodology often uses Markov Chain Monte Carlo (MCMC) simulations to overcome issues found in alternative statistical approaches, particularly in estimating probabilities and updating beliefs based on new data.