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And bonus

To be read when above theory is clear...

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And Bayesian?

Duck.ai...

Yes, there is a relationship between Bayesian methods and ordering, particularly in the context of Bayesian inference and decision-making.

1. **Bayesian Inference**: In Bayesian statistics, we update our beliefs about a parameter or hypothesis based on new evidence. This process involves calculating the posterior distribution, which is derived from the prior distribution and the likelihood of the observed data. The ordering of evidence can affect the posterior distribution, especially in cases where the evidence is sequentially observed.

2. **Ordering of Evidence**: In some Bayesian frameworks, the order in which data is observed can influence the inference process. For example, if you have a prior belief and you receive evidence in a specific order, the intermediate posteriors can differ based on that order. However, under certain conditions (e.g., when using conjugate priors), the final posterior distribution may be invariant to the order of evidence.

3. **Decision Theory**: In Bayesian decision theory, the ordering of possible outcomes and their associated probabilities can influence decision-making. Decision-makers often rank options based on expected utility, which is calculated using Bayesian principles. The ordering of these options can guide the choice that maximizes expected utility.

4. **Markov Chain Monte Carlo (MCMC)**: In computational Bayesian statistics, MCMC methods often rely on the ordering of samples to converge to the target distribution. The way samples are drawn and ordered can affect the efficiency and convergence of the algorithm.

In summary, while Bayesian methods are fundamentally about updating beliefs based on evidence, the ordering of that evidence and the resulting implications for inference and decision-making are important considerations in Bayesian analysis.

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Duck.ai...

Yes, there is a growing body of research suggesting that EEG (electroencephalogram) waves and brain activity can be understood through the lens of Bayesian principles, particularly in the context of predictive coding and Bayesian brain theories.

1. **Predictive Coding**: This theory posits that the brain continuously generates and updates predictions about sensory input. EEG studies have shown that brain waves, particularly in the alpha and beta frequency bands, reflect the brain's predictions and the precision of those predictions. When there is a mismatch between expected and actual sensory input (prediction error), the brain adjusts its predictions, which can be observed in the EEG signal.

2. **Bayesian Brain Hypothesis**: This hypothesis suggests that the brain operates as a Bayesian inference machine, integrating prior knowledge with incoming sensory information to make sense of the world. EEG studies have provided evidence that different frequency bands are associated with various cognitive processes that align with Bayesian inference, such as attention, perception, and decision-making.

3. **Event-Related Potentials (ERPs)**: Certain components of ERPs, which are derived from EEG data, have been linked to Bayesian processing. For example, the N400 component is associated with semantic processing and can reflect the brain's expectations about language and meaning, consistent with Bayesian principles of prediction and updating beliefs based on new information.

Overall, while the relationship between EEG waves and Bayesian principles is still an active area of research, there is substantial evidence supporting the idea that brain activity, as measured by EEG, can be interpreted through a Bayesian framework.

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History

Mervyn -> Rao 96 !!

Rao 98 publ ... Admin m. stood at or was resp

for the cradle of the Bayesian Brain

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The history of the future of the Bayesian brain

Karl Friston

https://www.sciencedirect.com/science/article/pii/S1053811911011657#bb0140

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