I believe it is essential to explain why it is independent in the case of the bathtub example and not in the other examples.
In the bathtub example, the evidence presents an event which is directly described by the assessed trait; i.e, the fairness of a coin is directly concerned with the appearance of either heads or tails. In contrast, the definition of the degree of "spamness" in an email is not directly concerned with the appearance of a word in the email, but is rather concerned with the abstract concept of the meaning a person assigns to the email.
The appearance of a word in an email is hence only an attempt of estimating the degree of "spamness", a proxy. In the case of a proxy, we need to consider the option that the proxy is flawed in a way which makes it so that the evidences are in fact dependencies of one another. This is not necessarily true, but it is possible, unlike in the case of hypothetical coins (in reality, a coin toss might actually be physically affected by the previous toss).
I find the entire explanation described below very misleading and perhaps even largely incorrect. The workshop participants had it wrong mostly for two reasons:
They did not consider what is the likelihood of visiting a museum / workplace given any other alternative (mutually exclusive) relationship - not strangers but also not romantically involved; i.e, friends. Being acquaintances is not a relevant type of a relationship as it is not mutually exclusive with a romantic relationship (a pair can be both dating and working together).
They did not know the prior probability of an arbitrary pair of people being romantically involved. A naive assumption of 50% of them being romantically involved is wrong, and should be made by observing the proportions of romantic relationships in the population.
In terms of the previous coins-fairness example, they (a) only considered that one type of coin (fair) is 2 times as likely to turn up heads as another type of coin (tail-biased), but did not consider how likely are the other type of coins (head-biased) to turn up heads; and (b) they did not know the proportions of coin types in the bathtub.
The explanation below also fails to mention the important assumption that the trait being assessed in all of the examples (coins, emails, workshop) is constant and doesn't change over time. It is important to mention because it may not be so trivial for every example, yet it reduces the complexity of the estimations tremendously. A coin is not expected to change its bias significantly over time, yet a relationship does, and so does the magnitude of "spamness" in a given mail for a given person (for instance, when I get older I may be more interested in pharmaceutical ads).
Actually, there should be diagonal matrices instead of vectors. Cross product doesn’t work like this, and dot product gives us a sum of coordinates of the vector we need instead of the vector itself, so we can’t continue updating our probabilities (or make any sense of the result). Diagonal matrices, on the other hand, do exactly what we need: C=AB;cii=aii∗bii;∀i≠j,cij=0.
What's the bathtub coins example? I've read the entire advanced sequence up to here and I don't remember reading about that. Maybe it was edited and removed? (Or maybe I wasn't paying attention or something?)
Are there going to be visual explanations put here for the examples? I found that quite helpful in the former pages. I'd say this is the first part of the new Bayes Guide that feels very similar (in terms of clarity) to the old one. Although, I might be biased as I've found I much prefer visual explanations of things.
It might be good to see an example worked out correctly; all we see here is an incorrect example.
I believe it is essential to explain why it is independent in the case of the bathtub example and not in the other examples.
In the bathtub example, the evidence presents an event which is directly described by the assessed trait; i.e, the fairness of a coin is directly concerned with the appearance of either heads or tails. In contrast, the definition of the degree of "spamness" in an email is not directly concerned with the appearance of a word in the email, but is rather concerned with the abstract concept of the meaning a person assigns to the email.
The appearance of a word in an email is hence only an attempt of estimating the degree of "spamness", a proxy. In the case of a proxy, we need to consider the option that the proxy is flawed in a way which makes it so that the evidences are in fact dependencies of one another. This is not necessarily true, but it is possible, unlike in the case of hypothetical coins (in reality, a coin toss might actually be physically affected by the previous toss).
I find the entire explanation described below very misleading and perhaps even largely incorrect. The workshop participants had it wrong mostly for two reasons:
They did not consider what is the likelihood of visiting a museum / workplace given any other alternative (mutually exclusive) relationship - not strangers but also not romantically involved; i.e, friends. Being acquaintances is not a relevant type of a relationship as it is not mutually exclusive with a romantic relationship (a pair can be both dating and working together).
They did not know the prior probability of an arbitrary pair of people being romantically involved. A naive assumption of 50% of them being romantically involved is wrong, and should be made by observing the proportions of romantic relationships in the population.
In terms of the previous coins-fairness example, they (a) only considered that one type of coin (fair) is 2 times as likely to turn up heads as another type of coin (tail-biased), but did not consider how likely are the other type of coins (head-biased) to turn up heads; and (b) they did not know the proportions of coin types in the bathtub.
The explanation below also fails to mention the important assumption that the trait being assessed in all of the examples (coins, emails, workshop) is constant and doesn't change over time. It is important to mention because it may not be so trivial for every example, yet it reduces the complexity of the estimations tremendously. A coin is not expected to change its bias significantly over time, yet a relationship does, and so does the magnitude of "spamness" in a given mail for a given person (for instance, when I get older I may be more interested in pharmaceutical ads).
Arguendo: more random non-common latin. Consider "For the sake of argument" or "Perhaps"
Actually, there should be diagonal matrices instead of vectors. Cross product doesn’t work like this, and dot product gives us a sum of coordinates of the vector we need instead of the vector itself, so we can’t continue updating our probabilities (or make any sense of the result). Diagonal matrices, on the other hand, do exactly what we need: C=AB;cii=aii∗bii;∀i≠j,cij=0.
I believe it should be, "the two were not romantically attracted" as that is consistent with the formula below.
I believe that this should be (2:3:1) rather than (3:2:1).
What's the bathtub coins example? I've read the entire advanced sequence up to here and I don't remember reading about that. Maybe it was edited and removed? (Or maybe I wasn't paying attention or something?)
Where did the '16' come from in (12/16:3/16:1/16) ?
It's 12 + 3 + 1. I'll edit to make clearer, but your comment exposed a bug in our LaTeX parsing so I'm waiting to edit until that resolves. :)
Are there going to be visual explanations put here for the examples? I found that quite helpful in the former pages. I'd say this is the first part of the new Bayes Guide that feels very similar (in terms of clarity) to the old one. Although, I might be biased as I've found I much prefer visual explanations of things.