How Probable Is The Probability?

Imagine one day you wake up with a bad feeling of not doing well, no major problems but just not feeling the same way how you would feel on any other day. You seek a doctor and he says that everything is absolutely fine, but since you were feeling unwell so he asks you to get a few tests so that he may get the idea if anything is causing you ill from inside and isn’t showing any visible symptoms. You get tested, and after a few days when the reports came back, the doctor finds something wrong and tells you that you have been detected with a very rare disease (just 0.1% of the world’s population suffers from it) which is really dangerous and makes all the stuff go wrong inside the body. You then ask the doctor about the authenticity of the test and the doctor assures with 99% probability of it being correct. Now, you would think that there is a 99% chance of you having that disease. But that is not the case. To know why, you have to dig into one of the excellent theorems of probability – The Bayes’ Theorem.

• The Bayes’ Theorem Simplified

➢Posterior

This tells you the probability of you having the disease after getting tested positive. The “A” signifies that you actually have the disease, given that you are tested positive for it which is signified by “B”.

➢Prior

This is the probability of you thinking to be tested positive before even getting test results, i.e. how likely you thought you would have the disease.

➢Likelihood

This is the probability of getting tests results as positive if you were having the disease.

➢Marginalization

The probability of the event occurring (i.e. testing positive). This term is actually a combination of 1.) Having the disease and correctly identified and 2.) Not having the disease and falsely identified.

The hardest part to figure is the “Prior” , and it is many times just a guess. But in your case the Prior can be taken as the frequency of the disease, i.e. 0.001  (0.1% as assumed previously). Now after plugging all other details:

We observe that we have a probability of 9% of having the disease even after testing positive for the same; this incredibly strange, isn’t it? This is a low figure as compared to what we presumed to be 99%! You might think this as some stupid concept that switches things like the way a magic whip does. But frankly speaking, this is just general common sense that is applied to mathematics.

For example, take a group of 1000 people. Let us suppose, one among them will have the disease and the test will identify it correctly. But the twist is that for the sample size of 1000 people we have taken, out of the other 999 people, to make up the 1%, ten (1% of 999 people is 10 people) will be falsely detected as positive! (Where 1% is the frequency of this disease)

So, from everyone who has a positive test result and you are selected at random, you are to be chosen from the group of 11 people (10 false + you). This brings us up to your chances of having the disease as 1 in 11, and that is 9%. This is real, isn’t it.

When Bayes came up with this, he didn’t think that it was a revolutionary thought and that it wasn’t worth publishing. Later, after his demise, the family members requested his juniors or assistants to go through his papers and check if anything worth publishing is left in these papers of brilliance. Believe it or not, there they found in his desolated papers this theorem of sheer excellence.

How Did Bayes come up with this idea?

Bayes used to sit with his back towards a square table, then asking one of his assistants to throw one ball on the table and he tried to guess the exact location of every ball. After the very first ball, he asked assistant whether the ball landed to the left or right, or to the front or behind of the very first ball; he kept continuing this as he jotted down everything. Hence, updating the location after each ball was thrown, trying to be more precise about the location of the first ball each time.

Bayes came up with the idea that this world could not be predicted up to utter precision but rather just could be updated time and again (say after every ball was thrown) to be closer to precision. When his juniors published this, he tried to explain the analogy with the example of a cave man- when the cave man saw the sunlight for the very first time, he might have thought of it as a quirk or an anomaly, but as the days passed, he would have realized how the Sun or this Earth works (by summing up the continuous evidences).

Bayes’ formula is not a one-time formula, but it is to be applied every time when we find out a new evidence. Now think of the very first illustration- now you go to a different doctor for checkup and a different lab to get tested. Guess What? you tested positive again! so now you will use the Bayes’ Formula once again to find out the actual probability, but with a bit of change. You will have to change the Prior part of formula because you already have one positive test.

After calculating these figures:

The final result comes out to be a 91%. i.e. there is a 91% chance of you being infected and getting the correct test report. But still the probability isn’t as high as the reported accuracy of the tests.

Thinking Of Practical Applications? Here Is The Answer

The Bayes’ theorem is used to create a modern filter or more precisely a spam filter which is also coined as “Bayes’ filter”.

It filters out an email as a spam or original and uses this above probability to check the viability of the email.

Bayes’ theorem tells us about how to upgrade with the new evidences and keep tweaking to move forward. Imagine a person with the 100% certainty about an outcome, while the other person having 0% certainty about the same outcome. According to Bayes, there is no sense of creating a debate between those two people on that specific topic as there is no possible room for a change and we won’t be able to come at a decision or won’t even encounter chance of one convincing the other of anything.

People think that how weakly intuitive they are about the analogy of Bayes’ theorem, but the reality is that people live in a strongly Bayesian world where anyone who faces continuous rejections, low wages or continuous descend builds up his or her intuition about it being certain as it had happened enough amount of times. Hence, you would not have any “Prior” (often termed as being hopeless) for that event and this will make it impossible for the event to happen.

What many usually miss is that how crucial their play is in determining the outcome of a thing. So, if we keep on thinking that something is 100% “true” or 100% “false” then we won’t be able to bring an about change for the same as no matter how many evidences we get, there will be no cumulative effect on that “100%”.

Bayes’ theorem suggests the idea of making changes. If you are performing an experiment for a long time and you aren’t seeing any favorable result, it’s high time to change! (Trying in a similar manner won’t increase the probability of it being possible)

Conclusion

› Bayes’ Theorem completely revolutionizes the vey basic concept of probability.

› Probability of any event heavily depends on how the person is performing it.

› New evidences may exceptionally change the probability outcomes.

› There can practically be nothing done to change a straight 100% or a 0%.

›  Every new evidence suggests about the progress of the experiment.

› Methods must be changed to obtain different outcomes (different than the ones which have been used for a long time now).

Report by Sashit Vijay