Evidence and counterexample to positive relevance

What do you think?

I think philosophers need to spend less time trying to come up with necessary-and-sufficient definitions of english words.

Your interpretation about g is correct.

The high probability interpretation is not a useful interpretation of "evidence," and there's a much easier way to discuss why: implication. P("A or ~A"|"My socks are white")=1, because P("A or ~A") is 1, and conditioning on my socks being white cannot make that less true. It is not sensible to describe the color of my socks as evidence for the truth value of "A or ~A".

The increase in probability definition is sensible, and what is used locally for Bayesian evidence.

Have you read this?

Um, sorry, but seriously?! Arguing about definitions of words? This is entirely ridiculous and way below the minimum rationality that should be expected from posts on Less Wrong. Downvoted for proposing serious discussion of a topic that deserves no such thing. Since you seem sincere I'll try and give you a quick overview of the problems here, but you really need to reread the sequence "A Human's Guide to Words" to get a full picture.

First, while I have an answer to what the useful definition of evidence is (in the sense that it describes a useful feature of reality), I will refrain from pointing it out here because it is irrelevant to the topic at hand. If someone really needed the word "evidence" for some reason, including potential hypothesis-favoring-data sufficient to convince me that most people mean something very different from me by the word "evidence", I'd be willing to give up the word. After all, words don't have meanings, they're just mental paintbrush handles for someone else's brain, and if that handle paints the wrong picture then I'll use a different one.

That said, the thrust of the problem with your post is exactly the same as the definitional dispute over a tree in a forest. There is no "true" meaning of evidence, and anyone arguing about one is doing so with an intent to sneak in connotations to win an argument by appealing to basic human fallacies. Definitional Disputes are an indisputably Dark Side tactic; the person doing might be honest, but if so then they are severely confused. Most people couldn't identify the difference between good and bad epistemology if it hit them in the face, and this does not make them evil, but it does make them wrong. Why would anyone care what the "true meaning" of evidence is, when they could just break down the various definitions and use each consistently and clearly? The only reason to care runs along the lines of "evidence is an important concept [hidden inference], this is the true definition, therefore this definition is important", replacing "important" with something specific to some discussion.

Only think about words as paintbrush handles, and the problem goes away. You can then start focusing on the concept behind your handle, and trying to communicate instead of win. Once you and your audience can all understand what is being said - that is, when the pictures you draw in their brain match the pictures in your head - then you're done. If you dispute anything at that point, it will be your true dispute, and it will have a resolution - either you have different priors, one or more of you is irrational, or you will walk away in agreement (or you'll run out of time - humans aren't ideal reasoners after all). Play Rationalist Taboo - what question is this post even asking, when you remove explicit reference to the word "evidence"? You can't ask questions about a concept which you can't even identify.

I feel like I've seen an increasing amount of classical Traditional Rationality bullshit on this site lately, mostly in Discussion. That could just be me starting to notice it more, but I feel like I need to make a full post about it that I can link to whenever this stuff comes up. These are basic errors, explicitly warned against in the Sequences, and the whole point of Less Wrong is supposed to be somewhere where this sort of crap is avoided. Apologies for language.

In the counterexample, e1 and e2 are each evidence of h on there own, but when the other is known, they are not.

(High Probability) e is evidence that h iff P(h|e) > k.

By this definition, if e and h are independent, but h has a prior probability higher than k, then e is evidence for h. For that matter, you could get something like P(h|e) = k+ε, P(h) = 1-ε. By this definition, e is evidence of h, even though e makes h dramatically less likely.

Also, this means that a confession is not evidence for a crime, because you need to know the language for it to mean anything.

I concur. Now consider the case where you observed the ticket sales, and you saw Bill buy 999 tickets (g). Then someone tells you that Bill bought 999 tickets(e1). Is e1 evidence that Bill will win the lottery?

Suppose that you saw the ticket sales, and saw someone other than Bill buy 501 out of 1000 tickets. Is there any possible evidence that Bill will win? (Assume tickets are nontransferable & other loopholes are accounted for-the odds are 501:499 against)

Suppose that you didn't see the ticket sales, but a reliable source (The New York Times) reports that Bill bought somewhere around 500 tickets, but they don't know the exact number. Would that be evidence that Bill will win? (Assume that there are many people as eligible to buy tickets)

I will continue using the definition "e is evidence of h iff P(h|e) > P(h)". I don't think that P(h|b) is meaningful unless P(h|~b) is also meaningful.

I think your example eviscerates the first "Increase in Probability" definition, at least as presented here, and shows that it doesn't account for non-independent evidence. If I wake up one morning and read in the New York Times reports that Bill Clinton has bought 99.9 % of tickets in a lottery, this strongly increases my estimate of the probability that Clinton will win the lottery. Reading the same story in the Washington Post does not materially increase my estimate given the background information in the New York Times. (I suppose it slightly reduces the probability that the Times story is a hoax or mistaken. Just maybe that's relevant.) Thus by this definition the Post story is not evidence (or at least is very weak evidence) that Bill Clinton will win the lottery.

However, suppose instead I wake up and read the Post story first. This now provides strong evidence that Bill Clinton will the lottery. The Times story is weak evidence at best. So depending on the irrelevant detail of which story I read first, one is strong evidence and one is weak evidence? That seems wrong. I don't want the strength of evidence to depend more on the irrelevant detail of which order I encounter two pieces of evidence. So perhaps what's being defined here is not the quality of the evidence but the usefulness of new evidence to me given what I already know?

Of course evidence and probabilities, especially non-independent probabilities are not additive.

This is not inobvious, so I notice that I am confused. I have to think that I'm misunderstanding this definition; or that there are details in the book that you're not reporting.

Memorizing a book might make you more knowledgeable, but memorizing the same book twice will just make you more bored.

Similarly, acquiring new information about the world will improve your probability estimates, but acquiring the same information in two different ways will just make your equations more complicated.

It seems very strange to me to look at e1 and e2 as point values without weights or confidence distributions. Taking no other information about e1 and e2 other than the fact that they both indicate a 999/1000 victory is quite limiting and makes for impossible meta-analysis. Other information could include what you think of both sources.

If you could get an accurate 90% confidence interval for each (or any accurate probability distribution) this could make a lot more sense. This must encompass the expected error in the New York Time's error margin, especially if they don't have one. For example, you may find that whenever reference a statistic, 90% of the time it is has at most 15% error compared to the true result (this would be really useful btw, someone should do this). Even if you estimate this number, you could still get a workable value.

If their 90% confidence interval was 0% error, and their reported statistics were always exactly true, then I do not believe you would update at all from e2.

I feel like it is possible to combine two 90% confidence intervals, and my guess is that any two with the same mean would result in higher certainty than at least the worst estimate (the one with the wider 90% confidence interval), possible higher than both. Solving this mathematically is something I'm not too sure about.