Probability is one of the tools that the human mind uses to try and make sense of this world. We use it all the time in our day to day activities for our survival and it has evolved over time (A Short History of Probability). As it evolved from being a subjective term used in philosophical literature to having quantitative forms, it started getting interpreted in a variety of ways (Probability Interpretations). One of its interpretations is the Bayesian reasoning.
Bayesian probability depends on previous knowledge to come up with the probability that an event can occur. Bayesian reasoning has been used in a variety of fields and especially in the field of artificial intelligence and machine learning.
A Simple Example
My friend Anna bakes either brownies or cupcakes every Sunday. I love brownies and I want to find out the probability of Anna baking brownies this Sunday. I can figure this out based on my previous experience of having brunch with her.
About 70% of the time she bakes brownies and she bakes cupcakes 30% of the time. I also know that when she has had a busy week, she finds it easier to bake brownies. Whenever she is busy she bakes brownies 80% of the time. And I know the past week she was busy.
With this information in hand, how do we find the probability of me having brownies this Sunday?
Just by seeing this information, without calculating mathematically, I think there is a good chance of me having brownies this Sunday. After the delicious meal Anna has cooked for us, I am all set to have my favorite brownie.
Drumrolls!!!!
She baked cupcakes.
How could this happen? There was a good chance she baked brownies right?
Or was there?
Let’s be total nerds and calculate the probability to find out.
P(Baking Brownies | Busy Week) =?
P(Baking Brownies) = 0.7
0.7 is called the prior. It is the knowledge from the past.
P(Busy Week | Baking Brownies) = 0.8
0.8 is called evidence. It is based on previous data.
P(Baking Brownies | Busy Week) = (0.8 * 0.7 ) / ((0.7*0.8) + (0.3*0.2))
= 0.56/1.06
= 0.35
0.35 is called posterior. The resulting probability.
Therefore, after using conditional probability, we find out that even though it looks as if the odds that Anna might have baked brownies are good, we see there is only 35% chance that she will bake brownies.
Next week, before we go to Anna’s, we do the same calculation. We will use 35% as the prior instead of 70%. There will be a certain degree of variation of probability from each prior and posterior. Each time Anna bakes a dessert, the probability gets updated with the new information. The probability can either increase or decrease as new information gets incorporated.
However, the quality of the probability will depend on the quality of the data we have. What if our data regarding Anna baking brownies about 70% of the time was wrong? What if other than the information we have, there are some other factors that affect Anna’s decision to bake brownies? Let’s say Anna’s significant other is visiting and who just like me loves brownies. And in that case, she would bake brownies 100% of the time.
Therefore, while we use the Bayesian method to calculate the probability, subjectivity from multiple angles needs to be factored in.
Hypothesis and Beliefs
In the above example, we have some data to find the probability and update it each time. However, most of the times we would not have the data, or it would be highly unlikely that the event would occur again. In such cases, we can apply Bayesian reasoning using hypothesizes.
For example, I am going to a wedding this weekend. I am in my mid-thirties and unmarried and I am expecting at least some of my relatives to ask about my relationship status.
My hypothesis is – I am in my mid-thirties. I will get asked about my relationship status.
How do I asses my chances of getting asked about my relationship status? I check if the subset of relatives whom I expect to meet usually ask me or others this question, let us say around 75% of the time. Therefore, I can be sure about encountering the unpleasantness according to their typical behavior.
Now I go to this wedding and meet the same group of people and spend time with them. However, they do not inquire about my status. I later learn that they got information that I have other priorities on my mind now and not looking to get married. So, I decrease the probability of the group of people asking me this question soon.
However, think of another scenario where they do ask me about my relationship status and I lie that I am committed to stopping them from asking again. I am providing them with the wrong information and the outcomes following this event will be dramatically different from before. Now chances of them enquiring about my significant other are more than 85% the next time I see them. The wrong information might spread to more number of people increasing the probability even more.
As you might have noticed, these assertions are not quantitative and far from being iron clad. I cannot be a hundred percent sure or a hundred percent unsure of my beliefs. They are more beliefs than evidence. The beliefs vary upon addition of new information be it right or wrong. This is called thinking in greyscale and shade of grey varies depending on the beliefs.
Conclusion
To sum it all up, calculating Bayesian probability assesses the strength of our belief and the strength changes with the addition of new information. As I mentioned in the introduction, the probability is a tool we use and not a physical entity. The tool depends heavily on the input it gets from our senses and it is impossible that the evidence is a hundred percent accurate and has many imperfections.
Even though the Bayesian reasoning is affected by a fair amount of subjectivity, it is highly useful. And it is possible to get better at it by collecting good evidence and applying quantitative methods to check the probability or the strength of our beliefs.
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