Are our brains capable of complex algorithms?

Does the Base Rate Fallacy Imply We are not Bayesian?

Introduction

The purpose of this piece is to explore the question of whether the Base Rate Fallacy implies that the human mind is not Bayesian. In weighing different psychological, neurobiological, and statistical sources, this paper seeks to explore the extent to which the human mind does function in a Bayesian manner, when it fails to do so, and the implications that stem from those two instances. The question as to whether the mind has Bayesian functions is hotly debated among many professionals. There is a lack of conclusive support and evidence to confidently make any inferences, but in this paper prominent theories and conjectures are tied together to support the idea that the mind is indeed not Bayesian.

Bayes Theorem
Bayes theorem is a probability calculation of something occurring based on prior knowledge. Many theories are derived from Bayesian claiming to explain both objective and subjective truths. Objective probabilities claim to be an extension of truth while subjective probabilities are decisions made based off on one’s biases, experiences, and personal knowledge. Bayes theorem can be stated as:
P(A|B) =P(B|A)P(A)/P(B)
A and B represent events P(B) ≠0
P(A) and P(B) represent probabilities not observed in relation to one another.
P(A|B) probability of event A given that B is true
P(B|A) probability of event B given that A is true. [1]
Base Rate Fallacy Derived from Bayes Theorem
From this theorem it is quite simple to illustrate a common fallacy. This is known as the base rate fallacy. An example will be given to illustrate this point. Suppose a hypothetical situation in which 1 out of every 1000 people who use their phone in the dark will contract a form of eye cancer, thus 999 people who use their phone in the dark likely won't. Additionally, the rate of error for the cancer diagnosis is approximately 5%. The doctor tells the patient that they have eye cancer, what is the probability that the patient does have eye cancer? Most people would say 95% but this is the fallacy. This seems counterintuitive but can be explained by Bayes theorem.
Looking at Bayes theorem we can assign an A and B to each part of the hypothetical situation.
P(A|B) = The probability that the patient has cancer in the case that the doctor has diagnosed them with cancer.
P(B|A)= The probability that the doctor will diagnose the patient with cancer given that the patient has cancer.
P(B|~A)=The probability that the doctor will diagnose the patient with cancer given that the patient does not have cancer.
P(A)= The probability of cancer
P(~A)=The probability of not having cancer
Solving for P(B)= P(B|A)P(A)+P(B|~A)P(~A)
The true probability of one having eye cancer would be: ( (1)(.001))/((1.001)+(.05*.99)) making it approximately 2%. The fallacy is that many will take the latter information into account (5% error rate of diagnosis) and fail to consider previous information. It is a logical fallacy but on the surface, does not appear intuitive at all. This base rate fallacy can be applied to situations such as terrorist attacks, drunk driving, etc. to expose the flaws in human reasoning when determining the likelihood of an event occurring.

The Bayesian Mind
Bayesian is the human capability to instinctively comprehend and process probabilities. It has been proposed by many neuroscientists and psychologists that the neural structure, connection, and general circuitry is arranged in a Bayesian manner. Simple actions and activities engaged in by people utilize the most basic neurological functions. As stimuli is received, cortical structures and sets of neurons are activated in response to either an anticipation of action or the actual performed action. Recent studies have suggested that neurons code in probability distributions and obvious patterns resembling Bayesian inferences can be seen. Bayesian optimality refers to the concept that known information in addition to uncertain, but stimuli relevant to the task is synthesized to create a model of uncertainty and is utilized to make decisions. Essentially, a distribution of known possibilities is coded in the neural patterns to guide behavior. [2] Many psychophysical experiments have confirmed that the brain uses such probability distribution [3] but what is not understood well is the mechanism and process in which this occurs on the molecular and neural level.

Proposed Conclusion Pertaining to the Question
It seems that often one’s perception of probabilities is based off personal experiences, while if one was truly Bayesian, the human mind would consider all possibilities. Humans have some innate instincts, (such as looking at a tower of blocks and being able to quickly determine whether it will fall) but the mind does not appear to be in the same capacity as a computer algorithm that figures out every possibility. The human mind often uses shortcuts to arrive at conclusions. For example, there is person X and Y and they go to the grocery store looking for a can of olives. Person X will look at every single can on every single aisle while person Y looks for the aisle that contains items such as beans and vegetables that would likely contain olives and then begins to search.
Often mathematical models can be formed to fit the hypothesis of many theorems. There have been studies done (such as the ones cited above) which claim that their statistical models of the brain’s coding patterns strongly show that the mind is Bayesian. Opposing scholarship dismisses these studies as an invalid scientific approach and calls for further evidence. The studies rejecting the idea of the Bayesian mind are cited below, if one hopes to further their understanding of these opposing theories. [4] [5] Popular culture, and often read scientific journals seem to have popularized the idea of the Bayesian mind when there has been little proof to support such a theory.
The base rate fallacy is one of many examples that the human mind is not Bayesian. One may have instincts that would lead one to think human mind is Bayesian, but just because a bird has wings and can fly like a jet plane, does not mean it has the capability for jet propulsion.
Returning to the core aspect of the question: does the base rate fallacy IMPLY we are not Bayesian? One cannot claim it implies we are not Bayesian, because it is not a direct deductive inference that one can make (a ->b , a thus b). There are many factors that allow ones’ mind to process probabilities, based on these factors and the available scholarship, one could conclude that it is safe to say that this strongly suggests that we are indeed not Bayesian. The debate is not settled and further research needs to be done to explore the neural mechanism and circuits that code for statistical probabilities in the human mind. If studies can be shown that the mind codes significantly more for Bayesian probabilities than it does algorithmically, conclusions will need to be readjusted.

[1] Stuart, A.; Ord, K. (1994), Kendall's Advanced Theory of Statistics: Volume I—Distribution Theory, Edward Arnold, §8.7.
[2] Spiking networks for Bayesian inference and choice Wei Ji Ma1,3 , Jeffrey M Beck1 and Alexandre Pouget1,2
[3] Kording KP, Wolpert DM: Bayesian integration in sensorimotor learning. Nature 2004, 427:244-247.
[4] https://www.ncbi.nlm.nih.gov/pubmed/22545686
[5] https://www.ncbi.nlm.nih.gov/pubmed/21864419