# Implementation of Bayesian inference on cognitive science research 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.

• cogsci.stackexchange.com/questions/10721/… – BCLC May 23 '16 at 18:49
• @BCLC Your linked question is interesting but not relevant to this question. – Kenny Kim May 28 '16 at 17:11
• check out my stats SE profile? – BCLC May 29 '16 at 2:57
• @BCLC Thanks. I found the paper about Bayesian Hierarchical Logit model attached to one of your questions to be very interesting. I'll read more on it tonight. eflglobal.com/wp-content/uploads/2014/09/… Also, I didn't know that cross validated stack exchange was stats site. Maybe I'll ask this type of question there in the future. – Kenny Kim May 29 '16 at 3:48
• Kenny Kim, there's this too. I also made a (not so good) paper about it. If you wanna see it, post your email or something I guess. – BCLC May 29 '16 at 4:11