Decision making is one of the main tasks of our brain for allowing survival. Decision making is founded on the probability estimation, trying to predict the likelihood of the events occuring in the future. Who better predicts, he’ll have more chance to survive.
We have three way to consider probability:
- probability is a property of the event (laplacian approach)
- probability is the past frequency of the event (frequentist approach)
- probability is a belief of the observer about the occurrence of the event (bayesian approach)
Lets’ consider the action of rolling a die.
If we apply the laplacian approach, we are considering the probability a property of the object-event. Having the dice six faces with equal chances, the chance of get 1 is 1/6 (16,67%). Always according to laplacian theory, there is no connection between not rolling a 6 on the first dice roll, and getting a 6 on the next roll. The probability will be the same – 1/6. Each event is classed as being independent.
If we apply the frequentist approach, we are going to consider the past frequency of the event onto the future events. E.g., if in the last 1000 events we have got that 1 has turned 180 times, we assign 18% of chance we get 1 in the next future event. Because the past estimates the future each time you roll the dice, you consider having 18% of chance of getting 1. This is the main approach we apply in medical literature.
Let’s Imagine a Bayesian specialist observing a game of dice in a casino. It is more than likely that he will begin with the same 1 in 6 chance, or 16.67% (Laplacian approach). As the night wears on, he notices that the dice is turning up sixes more than expected, and adjusts his belief. He begins to suspect that the dice is loaded, so leaves, keeping his money in his pocket. Applying a bayesian approach, we are going to upgrade relentlessly the likelihood estimation of getting 1 according to the information we have before rolling the die (prior probability) and the information we get after rolling the die (posterior probability). At the first toss, we are going to estimate the likelihood of getting 1 by means of laplacian approach (the dice having 6 faces with equally chance…this is the only information we have at the beginning). After “n” rolls, we have got new informations. If we had 99 coin tosses with heads arising, you would would recalculate the odds would. It wouldn’t be a 50-50 chance any more. You will also consider that tossing the die is an human, who has some “tendency” in his hand……whose hand can fatigue changing the tendency….you cannot consider every event independent. So you need to upgrade the odds at every toss (combining prior probability with a posterior probability), updating your beliefs. Each future event is dependent on the observer’s beliefs.
Let’s do another more explicative example in the real life. Let’s imagine we want to estimate what is the likelihood that a couple of caucasians, Eric and Samantha, will give birth to a caucasian child.
If we use the Laplacian approach, we should assume that the probability is the property of “being caucasian”. Thus, the probability is 100% that the newborn will be caucasian, being the parents both caucasians.
If we use the frequentist approach, the probability is the past frequency of the event. So we have to control the epidemiological data among caucasian couples. If we find that 97% of the children born from caucasian couples were caucasian, we’ll give 97% of probability to the upcoming event.
If we use bayesian approach, the situation may change drammatically. In its most basic form, bayesian probability it is the measure of confidence, or belief, that a person holds in a proposition.
Let’s consider that Eric and Samantha have 3 friends:
They are asked to express their estimation of the probability that their caucasian friends, Eric and Samantha, will give birth to a caucasian child.
Marc has no additional informations, further than general statistics in his country. Marc is therefore going to apply the laplacian or frequentist approach, giving 97-100% of chance the newborn will be caucasian.
Mary, instead, knows that Samantha has an extramarital relation with a black man. She cannot help to weight this information upon the probabily estimation of the future event. She’ll decide do not rely only on the frequency of past events (frequentistic approach) or just on the fact that they are caucasian (laplacian approach). She’ll rely more on the information she has (her beliefs). And depending on how much strongly she trust on that information, she’ll give different estimate of probability (since how many years Samanha has an extramarital relation? Does Samantha use contracceptive pills? etc etc.)
John has even more informations. He knows that Samantha has an extraconiugal relation with a black man and Erik suffer of absolute azospermia. John is going to completely overlook the frequentist data (past events), and should give almost 0% if chance that newborn baby will be caucasian.
Medical literature and guidelines, recipes are mainly and heavily based on “frequentist approach”. They create an average “man” and gives the same probability to all the patients.
But a doctor in most of the cases cannot rely on the data of literature for treating the single patient, because the beliefs he owns makes the literature data useless, and even dangerous for decision making.
Let’s do an example. If a doctor for the first time visit a female of 40 years old with a lump at the breast, the chance he’ll give to the possibility of tumor is the frequency that a female of that age with lump has (1%)
But if another doctor that knows that patient since 10 years and know that the patient has a lump since 10 years, will give completely another estimate of likelihood of cancer.
If a dentists at the first examination of a patient sees a radiolucency on the periapical x-ray, his threshold of action will be different from a a dentist who has been following-up that radiolucency for 10 years.