(This post is not an attempt to convey anything new, but is instead just an attempt to provide background context on how  Bayes' theorem works by describing how it can be deduced. This is not meant to be a formal proof. There have been other elementary posts that have covered how to use Bayes’ theorem: here, here, here and here)

 

Consider the following example

Imagine that your friend has a bowl that contains cookies in two varieties: chocolate chip and white chip macadamia nut. You think to yourself: “Yum. I would really like a chocolate chip cookie”. So you reach for one, but before you can pull one out your friend lets you know that you can only pick one, that you cannot look into the bowl and that all the cookies are either fresh or stale. Your friend also tells you that there are 80 fresh cookies, 40 chocolate chip cookies, 15 stale white chip macadamia nut cookies and 100 cookies in total. What is the probability that you will pull out a fresh chocolate chip cookie?

 

To figure this out we will create a truth table. If we fill in the values that we do know, then we will end up with the below table. I have highlighted in yellow the cell that we want to find the value of.

 

If we look at the above table we can notice that, like in Sudoku, there are some values that we can fill in based on the information that we already know. These values are coloured in grey and they are:

• The number of stale cookies. We know that 80 cookies are fresh and that there are 100 cookies in total, so this means that there must be 20 stale cookies.
• The number of white chip macadamia nut cookies. We know that there are 40 chocolate chip cookies and 100 cookies in total, so this means that there must be 60 white chip macadamia nut cookies

 

If we fill in both these values we end up with the below table:

 

If we look at the table now, we can see that there are two more values that can be filled in. These values are coloured in grey and they are:

• The number of fresh white chip macadamia nut cookies. We know that there are 60 white chip macadamia nut cookies and that 15 of these are stale, so this means that there must be 45 fresh white chip macadamia nut cookies.
• The number of stale chocolate chip cookies. We know that there are 20 stale cookies and that 15 of these are white chip macadamia nut, so this means that there must be 5 stale chocolate chip cookies.

 

If we fill in both these values we end up with the below table:

 

We can now find out the number of fresh chocolate chip cookies. It is important to note that there are two ways in which we can do this. These two ways are called the inverse of each other (this will be used later):

• Using the filled in row values. We know that there are 80 fresh cookies and that 45 of these are white chip macadamia nut, so this means that there must be 35 fresh chocolate chip cookies.
• Using the filled in column values. We know that there are 40 chocolate chip cookies and the 5 of these are stale, so this means that there must be 35 fresh chocolate chip cookies.

 

 If we fill in the last value in the table we end up with the below table:

 

We can now find out the probability of choosing a fresh chocolate chip cookie by dividing the number of fresh chocolate chip cookies (35) by the total number of cookies (100). This is 35 / 100 which is 35%. We now have the probability of choosing a fresh chocolate chip cookie (35%).

 

To get to the Bayes' theorem I will need to reduce the terms to a simpler form.

• P(A) = probability of finding some observation A. You can think of this as the probability of the picked cookie being chocolate chip.
• P(B)  = the probability of finding some observation B. You can think of this as the probability of the picked cookie being fresh. Please note that A is what we want to find given B. If it was desired, then A could be fresh and B chocolate chip.
• P(~A) = negated version of finding some observation A. You can think of this as the probability of the picked cookie not being chocolate i.e. being a white chip macadamia nut instead.
• P(~B) = a negated version of finding some observation B. You can think of this as the probability of the picked cookie not being fresh i.e. being stale instead.
• P(A∩B) = probability of being both A and B. You can think of this as the probability of the picked cookie being fresh and chocolate chip.

 

Now, we will start getting a bit more complicated as we start moving into the basis of the Bayes’ Theorem. Let’s go through another example based on the original.

Let’s assume that before you pull out a cookie you notice that it is fresh. Can you then figure out the likelihood of it being chocolate chip before you pull it out? The answer is yes.

 

We will find this out using the table that we filled in previously. The important row is underlined.

 

Since we already know that the cookie is fresh, we can say that the likelihood of it being a chocolate chip cookie is equal to the number of fresh chocolate chip cookies (35) divided by the total number of fresh cookies (80). This is 35 / 80 which is 43.75%.

 

In a simpler form this is:

• P(A|B) - The probability of A given B. You can think of this as the probability of the picked cookie being chocolate chip if you already know that it is fresh.

 

Let’s now return to the inverse idea that was raised previously. If we want to know the probability of the picked cookie being fresh and chocolate chip, i.e. P(A∩B), then we can use the underlined parts of the filled in truth table.

	Chocolate Chip	White Chip Macadamia Nut	Total
Fresh	35	45	80
Stale	5	15	20
Total	40	60	100

If we know that the cookie is known to be fresh like in the top row above, then we can find out that: P(A∩B) = P(A|B) * P(B). This means that the probability of the picked cookie being fresh and chocolate chip (35 / 100)  (remember that there were 100 cookies in total) is equal to the probability of it being chocolate chip given that you know that it is fresh (35 / 80) times the probability of it being fresh (80 / 100) . So, we end up with P(A∩B) = (35 / 80) * (80 / 100) which becomes 35% which is the same as 35 / 100 which we know is the right answer.

 

Alternatively, since we know that we can convert P(A|B) to P(A∩B) / P(B) (we found this out previously) we can also find out that:P(A∩B) = P(A|B) * P(B). We can do this by using the following method:

1. Assume P(A∩B) = P(A|B) * P(B)
2. Convert P(A|B) to P(A∩B) / P(B) so we get P(A∩B) = (P(A∩B) * P(B)) / P(B).
3. Notice that P(B) is on both the top and bottom of the equation, which means that it can be crossed out
4. Cross out P(B) to give you P(A∩B) = P(A∩B)

 

The inverse situation is when you know that the cookie is chocolate chip like in the left column in the table above. Using the left column we can find out that:  P(A∩B) = P (B|A) * P(A). This means that the probability of the picked cookie being fresh and chocolate chip (35 / 100) is equal to the probability of it being fresh given that you know it is chocolate chip (35 / 40) times the probability of it being chocolate chip (40 / 100). This is: P(A∩B) = (35 / 40) * (40 / 100). This becomes 35% which we know is the right answer.

 

Now, we have enough information to deduce the simple form of Bayes’ Theorem.

Let’s first recount what we know:

1. P(A|B) = P(A∩B) / P(B)
2. P(A∩B) = P(B|A) * P(A)

By taking the first fact: P(A|B) = P(A∩B) / P(B) and using the second fact to convert P(A∩B) to P(B|A) * P(A) you end up with P(A|B) = (P(B|A) * P(A)) / P(B) which is Bayes' Theorem in its simple form.

 

From the simple form of Bayes' Theorem, there is one more conversion that we need to make to derive the explicit form of Bayes' Theorem which is the one we are trying to deduce.

 

To get to the explicit form version we need to first find out that P(B) = P(A) * P(B|A) + P(~A) * P(B|~A).

To do this let’s refer to the table again:

 

We can see that the probability that the picked cookie is fresh (80 / 100) is equal to the probability that it is fresh and chocolate chip (35 / 100) plus the probability that it is fresh and white chip macadamia nut (45 / 100). So, we can find out that the probability of P(B) (cookie is fresh) is equal to 35 / 100 + 45 / 100 which is 0.8 or 80% which we know is the answer. This gives the formula:P(B) = P(A∩B) + P(~A∩B)

 

We know that P(A∩B) = P(B|A) * P(A) as we found this out earlier. Similarly we can find out that  P(~A∩B) = P(~A) * P(B|~A). This means that the probability of the picked cookie being fresh and white chip macadamia nut (45 / 100) is equal to the probability of it being white chip macadamia nut (60 / 100) times the probability of it being fresh cookie given that you know that it is white chip macadamia nut (45 / 60). This is: (60 / 100) * (45 / 60) which is 45% which we know is the answer.

 

Using this information, we can now get to the explicit form of Bayes' Theorem:

1. We know the simple form of Bayes' Theorem: P(A|B) = (P(B|A) * P(A)) / P(B)
2. We can convert P(B) to P(A∩B) + P(~A∩B) to get P(A|B) = (P(B|A) * P(A)) / (P(A∩B) + P(~A∩B))
3. We can convert P(A∩B) to P(A) * P(B|A) to get P(A|B) = (P(B|A) * P(A)) / (P(A) * P(B|A) + P(~A∩B))
4. We can convert P(~A∩B) to P(~A) * P(B|~A) to get P(A|B) = (P(B|A) * P(A)) / (P(A) * P(B|A) + P(~A) * P(B|~A))

Congratulations we have now reached the explicit form of Bayes' Theorem:  

If people don't see you as being “reasonable”, then you are likely to have troublesome interactions with them. Therefore, it is often valuable to be seen as “reasonable”. Reasonableness is a general perception that is determined by the social context and norms. It includes, but is not limited to, being seen as fair, sensible and socially cooperative. In summary, we can describe it as being noticeably rational in socially acceptable ways. What is “reasonable” and what is rational often converges, but it is important to note that they can also diverge and be different. For example, it was deemed “unreasonable” to free African-Americans from slavery because slavery was deemed necessary for the economy of the South.

 

The just-be-reasonable predicament occurs when you are chastised for doing something that you believe to be more rational and/or optimal than the norm or what is expected or desired. The chastiser has either: not considered, cannot fathom or does not care that what you are doing or want to do might be more rational and/or optimal than what is the default course of action. The predicament is similar to the one described in lonely dissent in that you must choose between making what you to believe to be the most rational and/or optimal course of action and the one that will be meet with the least amount of social disapproval. 

 

An example of this predicament is when you are playing a game with a scrub (a player who is handicapped by self-imposed rules that the game knows nothing about). The scrub might criticise for continuing to use the best strategy that you are aware of, but that they thinks is cheap. If you try to argue that a strategy is a strategy, then the argument is likely to end with the scrub getting angry and saying the equivalent of “just be reasonable”, which basically means: “why can’t you just follow what I see as the rules and the way things should be done?” When you encounter this predicament, you need to weigh up the costs of leaving the way or choosing a non-optimal action vs. facing potential social disapproval. The way opposes being “reasonable” when it is not aligned with being rational. In the scrub situation, the main benefit of being “reasonable” is that you are less likely to annoy the scrub and the main cost is that you are giving up a way to improve for both you and the scrub. The scrub will never learn how to counter the “cheap” strategy and you won’t be looking for other strategies as you know you can always just fall back to the “cheap” strategy if you want to win.

 

In general, you have three choices for how to deal with this predicament: you can be “reasonable”, explain yourself or try to ignore it. Ignoring it means that you continue or go ahead with the ration/optimal course of action that you had planned and that you also to change the conversation or situation so that you don't continue getting chastised. Which choice you should make depends on thecorrigibility and state of mind of the person that you need to explain yourself to as well as how much being “reasonable” differs from being rational. If we reconsider the scrub situation, then we can think of times when you should, or at least most people would, avoid the so called “cheap” strategy. Maybe, it is a bug in the game or it’s overpowered or your goal is fun rather than becoming better at the game. (Note, though, that becoming better at a game often makes it more fun).

 

The just-be-reasonable predicament is especially troubling because, like with the counter man syndrome, repeated erroneous thinking can become embedded into how you reason. In this case, repeated acquiescence can lead to embedding irrational and/or non-optimal ways of thinking into your thought processes.

 

If you continually encounter the just-be-reasonable predicament, then it indicates that your values are out of alignment with the person that you are dealing with. That is, they don’t value rationality, but just want you to do things in the way that they expect and want. Trying to get them to adopt a more rational way of doing things will often be a hard task because it involves having to convince them that their current paradigm from which they are deriving their beliefs as to what is “reasonable” is non-optimal.

Situations involving this predicament come in four main varieties:

If you encounter the just-be-reasonable predicament, I recommend running through the below process:  

 

Some other types of this predicament would be “just do as you’re told”, “why can’t you just conform to my belief of what is the best course of action for you here” and any other type of social disapproval, implicit or explicit, that you get from doing what is rational or optimal rather than what is expected or the default.