Implementation of Bayesian inference on cognitive science research question

Question

To clarify, I am not talking about Bayesian model of cognition. That model refers to the theory that the human learning and inference approximately match Bayesian inference reasoning. (The two topics are not unrelated though).

What I am talking about is the choosing of one hypothesis over another in explaining sets of data obtained. I recently read this paper (Jefferys & Berger 1992) that applies Bayesian inference to quantitatively explain Ockham's razor (the rule of thumb that simpler hypotheses are better).

Here is the link to the article: http://www.jstor.org/stable/29774559

Bayesian inference description

In the paper, the author implemented Baye's theorem to calculate the probability of a hypothesis being correct when new data is presented. The equation is as follows:

Where $P(H_i|D\&I)$ is the probability of a certain hypothesis being true given prior information (I) and new data (D). $P(D|H_i\&I)$ is the probability of new data (D) occurring given prior information (I) and the hypothesis being true (Hi). $P(H_i|I)$ is the prior probability ascribed to the hypothesis (Hi). $P(D|I)$ is the probability of new data (D) occurring given prior information (I) no matter whether the hypothesis is true.

In the paper, the author applied the Bayesian inference to show how data given for each trial can lead to weighing of one hypothesis vs another.

The example he gave compared two hypotheses: 1st hypothesis (H1) states that the coin has one head and one tail. 2nd hypothesis (H2) states that the coin has two heads. Initially, probability of each hypothesis being true is likely (there has not been a coin flip, and the experimenter does not know the coin at all). Thus, $P(H_1|\emptyset) = P(H_2|\emptyset) = 0.5$.

1st trial occurs, and the coin lands head. The probability of each hypothesis is modified.

For 1st hypothesis (1 head 1 tail), $P(Head|H_1\&\emptyset) = 0.5$, $P(H_1|\emptyset)=0.5$, and $P(Head|\emptyset)=unknown$. Thus, $P(H_1|Head)=\frac{0.25}{P(Head|\emptyset)}$.

For 2nd hypothesis (2 heads), $P(Head|H_2\&\emptyset) = 1$, $P(H_2|\emptyset)=0.5$, and $P(Head|\emptyset)=unknown$. Thus, $P(H_2|Head)=\frac{0.5}{P(Head|\emptyset)}$.

$P(Head|\emptyset)$ is normalization factor, and we are only comparing two hypotheses. Thus, $P(H_1|Head) + P(H_2|Head) = 1$. Thus, $P(H_1|Head) = 1/3$ and $P(H_2|Head) = 2/3$.

Flipping the coin multiple times and getting head each time decreases $P(H_1|Head)$ while increasing $P(H_2|Head)$ to near 1.

Applications to cognitive science and specifically neuroimaging or EEG

I found this method of hypotheses weighting to be very insightful and was wondering if there were applications of Bayesian inference to cognitive science in determining one hypothesis over another. Examples on neuroimaging or EEG data processing would be preferred since I am interested in that field, but any other example would be a good starting point.

Answer 1

A good starting point is probably Alex Etz's Understanding Bayes blog post series. He also co-authored a related paper that should be a great starting point: How to become a Bayesian in eight easy steps: An annotated reading list (link is to pre-print).

Richard Morey's BayesFactor blog is great resource for understanding the Bayesian model comparison approach. He has authored many papers that are relevant, but the blog is the best starting point for learning about the approach.

For a really accessible textbook, John Kruschke's Doing Bayesian Data Analysis is an excellent and thorough treatment of the use of Bayesian parameter estimation in data analysis. This approach is a bit different from the model comparison approach, but the conceptual ideas are the same. A related paper that motivates the approach in the book is available here.

Answer 2

Bayesian model comparison is widely performed. Here's a recent invasive electrophysiology example on decision-making:

• Single-trial spike trains in parietal cortex reveal discrete steps during decision-making Science, 2015