When I ask, how many of you believed in Santa Claus or the tooth fairy for the larger part of your lives, I already know it will be all of you because as kids your malleable brains were still developing to assess the evidence. Here is when Reverend Thomas Bayes comes in with his ‘Bayes’ Theorem’.
Bayes Theorem
While his theorem remained unpublished during his lifetime, it became greatly beneficial in many areas. Bayes' theorem states that the probability of A given B is the same as the probability of B given A times the probability of A, divided by the probability of B. It sounds a bit mouthful but it allows us to determine the chance of certain events happening when we know the probabilities of other related events beforehand.
Ever wondered how the weather is forecasted? By using the Bayes theorem. To understand the formula better, let us get our hands dirty by doing some math. What is the first sign you would look for it to rain? The clouds. So, to find the probability of rain given the clouds we would need the probability of the cloud given the rain, the probability of rain, and the probability of the cloud. To get it more into perspective, say you are currently in a desert where it rains once every 100 days making its probability at 1%, the chance for there to be a cloud in the desert on a given day is at 10%, and say, 80% of all rainy days in the desert begin cloudily. The total probability is
P(Cloud)=(P (C|R) ⋅ P(R)) / P(C) =(0.8⋅ 0.01)/0.1=0.08.
Therefore, if there is a cloud in the desert, the chance of rain is 8%.
On a similar note, unicorns are unreal and we all know it because there is no concrete or reliable proof of the existence of unicorns and your brain will immediately filter out the possibility of you seeing an actual unicorn.
Bayesian Brain hypothesis
All this can be explained through the Bayesian brain hypothesis; there is a deep hidden structure reasoning our behavior, the causes of which connect to the very nature of life. Our brain has an internal model of the world and its laws. After receiving new information, our brain updates it into the internal model, this is known as Bayesian inference. The brain structures its perceptions by generating probabilistic models of the external world.
Bayesian inference can be termed as the process of forming beliefs about the roots of sense (sensory) data. Sensory data is the data that we are aware of in perceptual experience before acts of inferring or judging, i.e., the unconscious process of assessing evidence.
We all use Bayesian inference in decision-making about real-world problems in life. When we want to select a solution for a problem among several solutions, then we usually analyze the past data from each solution or we try to predict the outcome of each solution then the solution with a higher success probability is selected. This process is alike Bayesian thinking when the decisions are selected based on far ahead probabilities rationalized by the Bayesian formula.
Viewing the unknown
Our brain is forever evolving, being exposed to different kinds of uncertainties. One of these uncertainties is ambiguity. You encounter it while looking at illusions, figuring out the ‘correct way’ to view it. You will notice that while your way is like a few other people, it is not the only one. This is probably why people in art museums spend long minutes viewing abstract art because their minds are trying to make sense of it based on its internal model.
Necker’s cube is an instance of a two-view image. It is an ordinary cube, except when you try to figure out which face of the cube is in the front. Now, brace yourselves because things are going to get a wee bit technical.
The human brain is made up of 86 billion neurons with more than 100 trillion synapses. Neurons are the cells in the brain that transmit and receive signals to enable processes such as thought. These signals are transmitted throughout junctions called synapses by neurotransmitters. The signals can be excitatory (which tends to make the neuron fire) or inhibitory (which prevents the neuron from firing). As neurons are connected by synapses, which have a connection strength; the stronger the strength the bigger the response of the receiving neuron. The neuron adds the strengths of the incoming signals (excitatory as positive and inhibitory as negative) and fires only if the total is big enough.
Coming back to Necker’s cube, you can see it in two different orientations, hence, there are two nodes, each responding to one orientation at a time. The fact that your brain can see both these orientations, though at different times, is proof of how complicated yet achieving our brain is.
‘Where am I?’
Every now and then when I drive, I often ask myself ‘Where am I?’ Maybe because I do not use a navigation app. But the context I want to propose through this question is a little different; how does your brain know where you are? Of course, we cannot comment on this regarding the human brain but we can take reference from rats’ brains. A discovery in 2005 stated that their brains have neurons known as grid cells that model their location in space.
We don’t know exactly how the grid cells tell the rat where it is. The layers of the grid cells ‘compute’ the rat’s location by integrating tiny movements as it wanders. Mathematically, this process can be realized using vector calculations, in which the position of a moving object is determined by adding lots of small changes, each with its magnitude. This is also how sailors used to navigate. The conclusion obtained from the study was that grid cells are vital for vector-based navigation.
More generally, the human brain uses circuits to understand the outside world by constantly learning, conditioned by laws and beliefs. Our beliefs are not like files on a computer that can be deleted or traded quickly. They are more wired in and changing them is hard. Like our brain, philosophies, and concepts under the Bayesian hypothesis are evolving. In the words of Nate Silver, “Under Bayes' Theorem, no theory is perfect. Rather it is a work in progress, always subject to further refinement and testing.”
References
Stewart, Ian. Do Dice Play God?: The Mathematics of Uncertainty. Profile Books, 2020.
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