Ali's Journal

based on wave probability, if a coin is tossed N times (N>>1) and never lands the tail, the probability that in the next round, it lands the tail should be higher than 1/2 (at the limit, N -> inf, the probability tends to 1).

But standard probability theory is different because each experiment is independent and therefore the probability is always 1/2. These two approaches are vividly different. However, the rule of large numbers can be used to link these two approaches. If N>>1, then n_t ~ N/2 and n_h ~ N/2 (where n_t refers to the number of times the coin lands the tail). As N -> inf, n_t -> N/2.

So if for some large N, the coin never lands the tail, then the rule of the large number should somehow deform the space so that the probability that in the next rounds, the coin lands the tail is higher.

UPDATE 1:

In other words, there should be a symmetry that deforms space so that n_h and n_t converge to N/2 (as N -> inf). The wave probability therefore requires mechanisms for this purpose. This distinguishes the wave probability from the standard probability theory.

(From now on biology-related topics will be discussed in this section of the website too)

The critical part of perception is to break down the environment and then reconstruct it. There are a lot more elements in the environment than it’s possible to consider so there should be an input filter that filters out a portion of the outside environment. Input sensors (such as the eye, ear, etc) will then decompose those elements and then they will be reconstructed in the brain.

So the mechanism to generate the approximation of the environment is key. That is done by the wave probability. For example, when an object is viewed from different perspectives its shape will differ (for example a circle can be seen as an ellipse when viewed from a different perspective). However, when input sensors receive info about different aspects of that object, it will be reconstructed in the brain. Then the wave probability assigns probability to past memories. Some are very far apart from that object and are filtered out but some memories get higher probabilities. the person might be uncertain at first which memories to link to that object but then as more data gets in the probability curve gets better and better and more relevant memories will be attached to that object.

So the discussion of problem solving where a complicated problem would be broken down into its constituent elements and perception of the environment are both linked together. Also, they are both linked to wave probability. Wave probability is the filtering mechanism that is critical in the process of the perception of the environment. It is said that rigorous mathematical precision cannot be used in physics. But maybe that is because approximation is encoded into nature for example similar to uncertainty. When an environment is observed the point at infinity cuts down a portion of the space. For example, imagine an infinitely long path with trees on both sides. Eventually, trees shrink into a point and the entire space beyond that will all be accumulated in the point at infinity. If there was no point at infinity there would be no observation because the path would continue forever. It is essential that the point at infinity cuts down a portion of space and provides an approximation. This may be true about any observable phenomenon in the physical world.

UPDATE 1:

If the approximation is indeed encoded into nature then since probability is linked to observation, then uncertainty and approximation are connected topics. One is derived from the other.

UPDATE 2:

The brain is the area of expertise of my dad. He is busy with his own research but I will ask him to advice me on this article on family gatherings.

I will write the proposal for the research and send it to the physician I mentioned a few days ago. I will ask her to be the co-author of the article.

UPDATE 3:

This is the proposal I will send her. Most of it has been discussed before.

Most environments consist of so many different elements that it is practically impossible to consider all of them. For example, a moving object is more probable to be noticed in a room where all other objects are fixed. So humans’ senses may ignore other elements in the room and concentrate on that object. If that moving object is removed, other objects that were not noticed at first might become noticeable.
The same is true in social contexts. When a catastrophe occurs, people tend to weigh more on things related to that incident and ignore other less relevant incidents. Therefore it’s more probable that those less relevant incidents will not be considered compared to more relevant incidents.
So a type of probability assignment process is necessary to attach different weights to different elements in the environment. This process relies on learning. That is due to, for example, evolutionary reasons some past incidents are stored in the memory and retrieved in similar situations. So those memories assign different weights to different elements in the environment. Standard probability theory lacks such a learning process. In fact, in some situations, probability relies on learning but in some other situations, standard probability is totally detached from the concept of learning. For example, to discover the probability of something, the experiment should be repeated many many times. As the number of experiments gets bigger and bigger the precision of the probability distribution gets more and more accurate. This is a learning process. It means all previous outcomes should be recorded, and the probability distribution is produced based on that.
However, in other situations, learning and memory are completely detached from the standard probability. Consider this example:

There are N balls in a bag and of one the balls is distinct. Based on the standard probability, the probability of randomly picking that distinct ball is 1/N. But based on the standard probability each experiment is independent. That is if a ball is picked each time and is returned to the bag afterwards, the probability of the distinct ball being picked equals 1/N every time. In mathematical terms those experiments are independent. It means it doesn’t matter which balls have been picked in the previous experiments. That is the same as saying the process lacks memory. However, in the real world things are different. If one picks balls repeatedly and each time returns the balls into the bag, and never the distinct ball is picked, as time goes on, psychologically one assigns more and more weight to the distinct ball being picked in the next round. That is because in this experiment memory is included.
It will be explained in later writings that this new type of probability that includes memory (also called wave probability) is closely linked to quantum mechanics and how the universe functions on the quantum scale.
This article aims to study if the same ideas can be applied to how the brain assigns probability distributions to different elements in the environment and also different memories. In other words, whether weight assignment in the brain is based on the wave probability.

It was said yesterday that a wavepacket can be considered one wave plus fuzziness. Also, in a wavepacket, a wave is localized. The same mechanism can be used in probability and observation.

It was said yesterday that each experiment is one observation. Each observation is a localized version of the probability wave. So wavepacket is very suitable for observation in probability. It means to observe an outcome in an experiment, the probability wave should be blurred so that it turns into a localized wavepacket so one outcome can be observed.

A mathematician trying to construct a structure using OOP is comparable to a game designer who designs the environment and gameplay of a game. That mathematician sets up all necessary tools and designs the environment where that particular construct is contained. The same method will be used in developing wave probability. It means at first, scenarios should be written about how different elements will be in the structure and then trying to construct them using OOP. So this is the scenario phase of constructing the wave probability. Evidently, this construction and scenario is not unique.

To estimate the probability distribution of something (call it P), the experiment should be repeated several times. Denote N as the number of times that experiment is repeated. if N is small then the variance around the mean is large, when N is large the variance shrinks. Each experiment can be considered to be one observation. Several observations are then required to produce an exact image of that thing. Precision here refers to low probability. In fact probability and observation are opposite concepts. when something is observed there is no probability. When a coin is tossed, the probability it lands head is 1/2 but when the outcome is observed, it has certainly landed head or tail, no probability. In fact, when observation has not occurred the outcome is a superposition of head or tail but when the outcome is observed, the wave collapses into either head or tail. This is similar to quantum mechanics.

About the previous example about the distribution of P, each experiment is one observation. but each observation alone provides a vague image of P, which is why the variance of P is so large when N is small. To re-construct this using OOP this scenario can be used (again this scenario is not unique):

P is a n-d wave rotating around itself. Each time its image is captured (i.e. after each experiment – equivalently each time the wave is observed) a blurred image of the wave will be produced. That is similar to a wave packet. In this scenario a wave packet is not a combination of several waves rather it is a wave with an added fuzziness. When more and more images are taken, the combination of these blurred images produces a somehow accurate image of the 3d wave. More images (i.e. large N), produce less probability (i.e. less inaccuracy). In this scenario, waves come with intrinsic rotations and motions. that links wave probability to SR. Note that SR and probability both depend on the concept of observation.

UPDATE 1:

Spin in the wave probability is similar to the same phenomenon in quantum mechanics. Due to the connection between SR and the wave probability, it’s beneficial to express a constant velocity in the form of spin. An object moving with constant velocity retains its velocity forever. It’s better to express that as an intrinsic rotation (or spin). Then it’s easier to connect SR to the wave probability.

In quantum mechanics, fermions are based on the Pauli exclusion principle. In Probability theory, possible outcomes for something are complements and the sum of all probabilities equals 1. To illustrate that similar graphs to the ones used in the set theory can be applied.

When the wave functions of fermions do not overlap, the Pauli exclusion principle is not used for them (so for example different tosses of a coin wouldn’t make the wave inference effect explained yesterday). The same ideas can be extended to the probability theory. Different outcomes can be considered as fermions or bosons (based on the situation – if they are independent or not).

UPDATE 1:

Complement is a term used in set theory. However, in probability complements exclude each other so it’s better not to use that word in probability. I used the word to link it to set theory.

UPDATE 2:

The inability to observe plays a role in probability. When a box contains invisible objects (visibility means they cannot be sensed by human senses), they cannot be picked.

The point at inf is a vital part of observation, this is also true in probability. A coin is hypothetically tossed an inf number of times and that’s how the probability has been calculated.