In the combox of the last post, we have an interesting example in which it seems that some random person’s assertion in fact constitutes zero evidence. I’d like to analyze that example in more detail.
Here’s the scenario. You’re sitting in front of your computer. Some random guy instant messages you, saying, “You’re eating raspberries.” Does this claim constitute evidence that you are, in fact, eating raspberries? It may intuitively appear that it does not.
Consider the possible scenarios:
A. The assertion that you are eating raspberries (I’ll call this the IM) is uninfluenced by the state of your eating.
B. The IM is influenced by the state of your eating.
Let H be the hypothesis that you are eating raspberries. Bayes’ theorem gives us:
P(A)+P(B)=1
R (the evidential ratio, P(IM|H)/P(IM|~H) ) = [(P(IM)*P(A))+(P(IM|H)*P(B))] / [(P(IM)*P(A))+(P(IM|~H)*P(B))]
The formula breaks down like this because on possibility A, P(IM|H) = P(IM); that is, if we assume that the IM is uninfluenced by the state of your eating, then it is equally likely regardless of that state.
Scenario A obtains with high probability, say 0.90. The exact value doesn’t matter, it can be as high as we want as long as it is not 1, which would imply that it is absolutely impossible that the sender of the IM could have any knowledge of whether you’re eating—obviously there is some chance of this, however small it may be.
Notice, though, that the probability of A is irrelevant, because it occurs in both the numerator and the denominator of the fraction, and the only thing that matters is the ratio. Thus, in order for the IM to not affect our judgment of P(H), it is necessary that given scenario B
P(IM|H)=P(IM|~H)
That is, the sender must be as likely to send the IM knowing that you are eating raspberries as he is likely to send the IM knowing that you are not eating raspberries. But to send an IM asserting that you are eating raspberries, knowing that you are not, is lying. Thus, claiming that the IM does not increase P(H) is equivalent to asserting that the sender is as likely to lie about what he believes as tell the truth about what he believes. Clearly this is hardly ever the case, and certainly would be irrational to assume of an anonymous person. Therefore, the IM increases the probability that you are eating raspberries.
A pre-emptive response to objections: “lie about what he believes” is not the same as “be in error about whether you are eating raspberries.” If you think it is, you didn’t understand the argument.
Second pre-emptive response: If you want to object that the assertion can be influenced by the state in some other way than the sender knowing the state, you need to describe the precise mode of influence.
Postscript: I haven’t assigned numbers, but in reality such an IM would constitute quite strong evidence, not very slight evidence. This is because of the improbability of someone sending such a message in the first place.
A generic ‘anonymous person’ might not be as likely to lie as say the truth in this case. But this is no generic anonymous person, now. This is a person who told me I’m eating raspberries when I know full well I’m holding a piece of chocolate in my hand. That puts him in a pretty select group in my mind. Which, in turn, raises the likelihood for me that he’s a liar of one sort or another.
Wouldn’t you say?
1. The problem as specified is slightly different, since it does not specify whether or not you are actually eating raspberries, or chocolate, or nothing at all.
2. It is true that the IM might slightly increase the probability of the sender’s being a liar. But that’s a different question than the probability of you eating raspberries, which is judged on the antecedent information. The same IM can increase the probability both that you are eating raspberries and that the sender is a liar. It certainly will not mean that the sender is more likely to lie than tell the truth given that he knows the truth. And this for reason 3 following:
3. More importantly, even if you do know with high probability (it will never be 1, despite the fact that you seem to like to talk about this as though it were a real possibility) that you are eating chocolate, not raspberries (mm…raspberries and chocolate), what this mostly does is vastly increase the proportion of the total probability space constituted by scenario A, namely that the IM was simply uninfluenced by the reality of what you are or are not eating. But this section of the probability space has no bearing on the probability of you eating raspberries, as shown in the post.
But anyway, I don’t think it would be fruitful to comment further on this until the much more fundamental difficulty you’re having grasping the difference between conditional and simple probabilities in the previous combox gets hashed out.
1. The problem as specified is slightly different, since it does not specify whether or not you are actually eating raspberries, or chocolate, or nothing at all.
Well, whatever the case, I know what I’m eating, or not, as the case may be.
2. It is true that the IM might slightly increase the probability of the sender’s being a liar.
I’ll be the judge of how much I think someone’s a liar when they tell me I’m eating raspberries when I’m not.
The same IM can increase the probability both that you are eating raspberries and that the sender is a liar.
It could. But in any given case, I’ll be the judge of which I think outweighs the other. And in a real-world instance of this scenario, I’m pretty sure that ‘liar’ would outweigh ‘raspberries’.
It certainly will not mean that the sender is more likely to lie than tell the truth given that he knows the truth.
That is not certain at all and I think it does mean that, personally. Again: by saying what he’s said, this person has self-identified himself to me as a person who is capable of telling me I’m eating raspberries when I know I’m not. Knowing this about the person gives me great confidence indeed that he is a liar of some sort.
3. More importantly, even if you do know with high probability (it will never be 1, despite the fact that you seem to like to talk about this as though it were a real possibility) that you are eating chocolate,
I think the probability can be 1. I wonder if you understand measure theory.
what this mostly does is vastly increase the proportion of the total probability space constituted by scenario A, namely that the IM was simply uninfluenced by the reality of what you are or are not eating.
That too. It does both. I increase my posterior likelihood that the asserter was either (a) uninfluenced by what I’m eating (for example, the IM was just spam or an advertisement of some sort), or (b) a liar. Both increase.
But this section of the probability space has no bearing on the probability of you eating raspberries,
Sure it does. How can you say ‘no bearing’? Look at your own math again. You got to a fraction of the form (A+B)/(A+B’) where A was thought to be a big number. Then you focused on B vs B’ which is fine if all you care about is whether the fraction is less than or bigger than 1. But the size of A still matters! The larger A is, the closer the fraction is to 1, of course, and thus the more certain I have to be that B and B’ are different, and about the sign of that difference, in order to change my posterior the way you want.
P.S. Feel free to point out where and how you think I had difficulty ‘grasping the difference between conditional and simple probabilities’. Granted it’s been a while since I completed my math PhD but I do believe I recall the basics.