Baye’s Rule for Population Neuroscience
Analyzing Brain Circuits in Population Neuroscience: A case to be a Bayesian (https://arxiv.org/abs/1909.02527)
In class, we considered a simple herding experiment in which a group of students chose a marble from a bag one at a time, and then guessed out loud whether the bag was majority red or majority blue. Based on intuition, we concluded that students would rely heavily on the past guesses of classmates. If the first student sees blue, and the second student sees blue, they will guess majority blue, and each student after them is likely to guess blue, even if they see red. Obviously a student holding a red marble will not think that all the marbles are blue, but since most of her classmates are guessing majority blue, she will logically infer, given this information, that most of the marbles are blue, and disregard her own information.
We then applied Baye’s rule to this situation to create mathematical support (see lecture notes or textbook). Baye’s rule uses conditional probability to make mathematically reasonable predictions in situations where the outcome is uncertain. Although it has drawbacks, it can be highly useful in applications far beyond the marble experiment. For example, neuroscientists seeking more effective diagnosis methods must sift through vast amounts of data to arrive at conclusions. Knowing the probability of a certain diagnosis given the presence of a certain symptom can be highly useful. In this case, a symptom is like the marble the student sees, and a diagnosis is like their guess that the bag is majority red or majority blue. Of course, in neuroscience studies, the bag of marbles is massive and multi-colored, information transfer is more complicated, and the guesses are hopefully a bit more accurate.
In the paper linked above, Bzdok et al. discuss the utility of a Bayesian perspective for drawing conclusions from high volumes of brain data. There is a specific emphasis placed on Bayesian analysis strategies for research questions related to autism spectrum disorder. For example, the paper states that, “under the Bayesian paradigm, probabilities reflect degrees of belief in a given proposition (e.g. ‘there is a low probability that the amygdala will activate 100 times more or less in autism vs. health’).” So given test results in which the amygdala activated a certain number of times, one can mathematically predict the likelihood that that person has autism.
According to the paper, using Baye’s rule for analysis “requires the specification of a prior distribution, reflecting our beliefs about the model parameters before observing any data.” In this context, this could be prior data connecting certain symptoms to certain diagnoses, or even prior data connecting genetic predispositions to diagnoses. The purpose of this data is to reduce noise, serving as a guide for sifting through many dependent parameters. Introducing assumptions like these can be problematic, but it is pointed out that because the Bayesian method makes assumptions explicit, it is in many ways better than less transparent or systematic analysis methods.
Although the paper provides many examples, the strength of the Bayesian method is well reflected in reducing the confounding effect of sex on autism diagnosis. Males are much more likely to be diagnosed with autism than females, for a variety of reasons, but beyond this, males and females can display vastly different symptoms. A male with autism might display a completely opposite measurement on a neurological test. Sorting out what differences in neurological symptoms are due to sex, versus which are due to autism spectrum disorder, can be highly complex. Bayesian Hierarchical modeling can answer “which sex-, age- and motion-related components in functional connectivity couplings are related to autism-related model parameters with which magnitude and how certain can the investigator be about it.” On the simplest level, one can apply conditional probability multiple times to address the many factors influencing diagnosis for a certain patient.