Bayesian Reasoning
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You climb a tree. What are the chances you’ll end up on a specific branch?
If I know how likely you are to drink coffee and how likely you are to get a good grade on a test then measuring how often you drank coffee before a good test also tells me how likely drinking coffee will irprove your test grade in the future.
Bayesian statistics is all about making a prediction and updating it as you go, or about figuring out how likely something is given that something else is true or false.
For example, you have two coins in a bag. One is a regular coin with a heads and a tails. The other is a trick coin with two heads. You pull one coin out of the bag. How likely is it that you picked the real coin? Now, you flip it, and it comes up heads. *Now* how likely is it that you picked the real coin? Say you flip the same coin again and get heads again. Does the likelihood that you picked the real coin change?
Another way to think of it is like a game of Marco Polo. In case you’re not familiar, one person is blindfolded. They should “Marco” and the other person (or people) reply “Polo.” For now, let’s assume you and a friend are playing in a large room. You’re blindfolded, and your friend isn’t allowed to move. You stand in the centre of the room and shout. You hear your friend off to your left. But you’re not sure if they’re in front or behind you, or how far left. So you walk diagonally forwards to the left, half way to the corner of the room. Now you shout again. When your friend replies, you can tell if they’re still to your left (i.e. close to the wall) or if they’re now on your right (i.e. between where you are and the middle). You can also tell if they’re closer to you or further away, i.e. in the front of back half of the room. You keep moving about and shouting. Each time you build up a better picture of where your friend is. Each of these turns wouldn’t be very helpful by themselves, but together they give you a lot of information.
One of the main fields where Bayesian analysis gets used (whether consciously or not) is in medicine. Say there’s a really rare illness that maybe 1/100,000 people have. One of the symptoms is a cough. This means is you pick someone off the street at random, there’s only a 1/100,000 chance they have it. Now, you develop a cough. You go to the Drs and mention this condition. Is it likely that you have it? Probably not. It’s rare, and a cough is a common symptom of a lot of stuff. But you have a slightly higher than 1/100,000 chance. Now, let’s say you are ginger, and this illness is more common amongst ginger people. Well, the chances you’ve got it are higher. Also, there happens to be an outbreak in your area. It’s now beginning to look much more likely that you have this condition. So, each of these fact separately might make it seem unlikely that you had this illness. But together, we get a more accurate picture of what’s going on.
There are two “interpretations” of statistics.
One that sees probabilities as something thats inherent to an event (e.g. you roll a dice, chance to get a 6 is 1/6, thats inherent to rolling dice.)
The other reasoning sees probability as something depending on the observer – e.g you look a lt a cloud and guess that its 50/50 that it’ll rain tomorow.
Then you check some satelite footage, get good data, and find out that its actually 90%. Nothing you did changed whether it actually rains or not, just your information and thus the probabilities ascribed by you changed.
Bayesian reasoning refers to the latter. Bayes created a bunch of equations to deal with changing information and how they affect probabilities, that nowadays are extremely important in science and engineering.
Another way of thinking of at least some uses of it is to take an answer to the wrong question and turn it into an answer to the question you actually want an answer to. This often comes up with medical testing, where you can have a test that’s very accurate but most of the positive results are false.
Say you have a test for Zoidberg’s Disease that’s 95% accurate. This is the answer to the wrong question — that means that if you have Zoidberg’s Disease, there’s a 95% chance that the test will be positive, and if you don’t have it, there’s a 95% chance the test will be negative. This isn’t any help to you, the patient, because if you knew you have Zoidberg’s Disease you wouldn’t go getting a test for it. What you want to know is: my test came back positive, what does that mean for me? And the answer will depend on how common Zoidberg’s Disease is.
Let’s say it’s rare, only 2% of people have it. So in our city of 100,000 people:
98000 don’t gots it
2000 gots it
And our test is 95% accurate, so:
* 95% of the 98000, or 91300, don’t gots it and get a negative result
* 4900 out of the 98000 don’t gots it but get a positive result
* 95% of the 2000, or 1900, gots it and get a positive result
* 100 of the 2000 gots it and get a negative result
Put that together, and how many people get a positive result? 6800. But only 1900 of them actually have Zoidberg’s Disease, so only 28% of positive results are correct even though 95% of total test outcomes are correct.
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