Evidence

Transitivity of Evidence Revisited

entirelyuseless Probability

It was earlier shown that evidence in general is not transitive. The reason given was that if A is evidence for B, and B for C, A is not necessarily evidence for C, because the cases where A and B are the case can be disjoint from the cases where B and C are the case.

However, this does not prevent evidence from being transitive on particular occasions, and possibly on most occasions, namely whenever situations where A and B are the case are the same or mostly the same situations where B and C are the case.

For example, if someone speaks with me in a loud tone and with a red face, that is evidence that he is angry with me; and if he is angry with me, that is evidence that I have said or done something which offended him; and likewise speaking with me in a loud tone and with a red face is evidence that I have said or done something which offended him. In this case situations where A and B are the case, namely where someone is angry and speaking angrily with me, overlap a good deal with cases where B and C are the case, namely where the person is angry because I have offended him.

This is quite possibly the more common situation — namely that evidence is transitive in practice — and thus the likely reason why people tend mistakenly to conclude that evidence is always transitive.

Simplicity and Probability

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Given some reasonable postulates regarding the formulation of explanatory hypotheses, it can be mathematically demonstrated that a probability distribution over all possible explanations will be biased toward simpler explanations — in an overall way the simpler explanations will be more probable than the more complex ones, although there may be individual exceptions.

We make the following postulates:

1) The explanatory hypotheses are described by a language that has a finite number of different words, and each hypothesis is expressed by a finite number of these words. That this allows for natural languages such as English, but also for computer programming languages and so on. The proof in this post will be valid for all cases. This is a reasonable assumption since human beings do not use any infinite languages, nor do they use an infinite number of words to make a point.

2) A complexity measure is assigned to the hypotheses in such a way that there are or may be some hypotheses which are as simple as possible, and these are assigned the complexity measure of 1, while hypotheses considered to be more complex are assigned higher integer values such as 2, 3, 4, and so on. Note that apart from this, we can define the complexity measure in any way we like, for example as the number of words used by the hypothesis, or in another way, as the shortest program which can output the hypothesis in a given programming language. Many other definitions would be possible. The proof is valid for all definitions that follow the conditions laid out, even the ones which would be intuitively somewhat distant from the idea of something simple. This again is a reasonable assumption given what we mean by simplicity — we do not think it is possible to make a thing infinitely simpler, but there is always something simplest.

3) The complexity measure is also defined in such a way that there are a finite number of hypotheses given the measure of 1, a finite number given the measure of 2, a finite number given the measure of 3, and so on. Note that this condition is not difficult to satisfy; it would be satisfied by either of the definitions mentioned in condition 2, and in fact by any reasonable definition of simplicity and complexity. If there are an infinite number of hypotheses that are supposedly absolutely simple (with the measure of 1), and we describe these hypotheses in English, an infinite number of them will not be able to be described without using at least 10,000 words, or without using at least 100,000 words, and so on. This seems very remote from the idea of a simple explanation.

With these three conditions the proof follows of necessity. To explain any data, in general there will be infinitely many mutually exclusive hypotheses which could fit the data. Suppose we assign prior probabilities for all of these hypotheses. Given condition 3, it will be possible to find the average probability for hypotheses of complexity 1 (call it x1), the average probability for hypotheses of complexity 2 (call it x2), the average probability for hypotheses of complexity 3 (call it x3), and so on. Now consider the infinite sum “x1 + x2 + x3…” Since all of these values are positive (and non-zero, since we consider each hypothesis to be at least possible), either the sum converges to a positive value, or it diverges to positive infinity. In fact, it will converge to a value less than 1, since if we had multiplied each term of the series by the number of hypotheses with the corresponding complexity, it would have converged to exactly 1, since the sum of all the probabilities of all our mutually exclusive hypotheses should be exactly 1.

Now, x1 is a finite real number. So in order for this series to converge, there must be only a finite number of terms in the series equal to or greater than x1, and therefore some last term which is equal to or greater than x1. There will therefore be some complexity value, y1, such that all hypotheses with a complexity value greater than y1 have an average probability of less than x1 (the average being taken over the hypotheses with the same complexity value, as above). Likewise for x2: there will be some complexity value y2 such that all hypotheses with a complexity value greater than y2 have an average probability of less than x2. Leaving the derivation for the reader, it would also follow that there is some complexity value z1 such that all hypotheses with a complexity value greater than z1 have a lower probability than any hypothesis with a complexity value of 1, some other complexity value z2 such that all hypotheses with a complexity value greater than z2 have a lower probability than any hypothesis of complexity values 1 or 2, and so on.

From this it is clear that as the complexity tends to infinity, the probability of the hypothesis will tend toward zero in the limit. This will happen in such a way that for any particular probability (e.g. one in a billion), there will be some degree of complexity such that every hypothesis at least that complex will be less probable than the chosen probability (e.g. less than one in a billion.)

Evidence and Implication

entirelyuseless Logic, Probability

Evidence and logical implication can be compared; we can say that logical implication is conclusive evidence, or that evidence is a sort of weak implication.

Evidence is commutative. If A is evidence for B, B is evidence for A. But logical implication is not; if A implies B, B does not necessarily imply A. However, even in the case of implication we can say that if A implies B, B is evidence for A.

Implication is transitive. If A implies B, and B implies C, then A implies C. We might be tempted to think that evidence will be transitive as well, so that if A is evidence for B, and B is evidence for C, A will be evidence for C. But this is not necessarily the case; this sort of thinking can lead to believing that the evidence can change sides. Attempting to make evidence transitive is like trying to draw a conclusion from a syllogism without any universal terms; if A is evidence for B, then some B cases are A cases, but not necessarily all of them; if every B case were an A case, then A would imply B, not merely be evidence for it. So if A is evidence for B, and B for C, then some C cases are B cases, and some B cases are A cases; but we cannot conclude that any C cases are A cases. A and C may very well be entirely disjoint. Thus the theory of evolution, taken as given, is evidence that transitional fossils between man and ape can be found; and the finding of such a transitional fossil is evidence for the (completely implausible) theory that some fossils have been preserved from every kind of organism that has ever inhabited the earth. But the theory of evolution taken as given does not provide evidence for the completely implausible theory; rather, the theory of evolution and the completely implausible theory would refute one another, at least if we are also given a little bit of background knowledge.

Using Arguments to Arrive at Understanding

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As I suggested in the previous post, we come to understand something better through a number of arguments rather than through a single argument.

Suppose you prove your position through a single argument that seems strong to you. In this case there is a second-order consideration which significantly weakens your argument. Namely, if you already suspected or held a position, or if you wanted it to be true or to believe it, how likely is it that you would manage to find at least one argument in favor of that position which seemed strong to you, given that the position was false? It is probably not much less likely than the same thing given that the position is true, and so the strong argument should not increase your belief in that position by very much. This of course does not imply that you should ignore the content of the argument, but it does mean that you should take it with a bit of caution. Approaching the matter with many arguments weakens this second-order consideration and gives you more reason to accept the implications of the arguments.

Using a number of arguments also helps you to refine your view, making it more precise, giving you a better ability to resolve objections, and so on. This is certainly one of Aristotle’s reasons for proposing the use of dialectic in coming to understand, and a reason for the use of many arguments in disputed questions, as I said in the previous post.

On the other hand, even if you come up with multiple arguments for your position, this may not be very helpful if you ignore opposing evidence, and so it is necessary to construct arguments against your position as well. This is the reason that a disputed question has arguments on both sides.

If you manage to construct a large number of arguments on both sides of a position, this will often give you a very strong basis for judging the truth of the position. It is difficult to assign numerical probabilities, and consequently to determine the exact strength of the evidence or of an argument for a position. But it is often comparatively easy to see the relative strength of two pieces of opposing evidence, or two opposing arguments. Consequently once such a list of opposing arguments has been constructed, it is possible to look at one side and see how the arguments compare to those for the other side.

As I have said earlier, there is evidence for any position, whether it is true or not. However, the evidence for a false position generally tends to be weaker than the evidence for a true position. So for example if nearly all the arguments for one side of a position are fairly weak, while many of the arguments for the other side seem significantly stronger, we can get a pretty good sense of which position is true and which is not.

On another note, there is a good post against the Equality Dogma here.

Confirmation Bias

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Confirmation bias is related to the tendency to say that all of the evidence is on my side. In particular it consists in attending to evidence that supports my position while ignoring contrary evidence, or interpreting the contrary evidence so that it appears to be supporting evidence. One way to resist this tendency is to notice that despite the saying, absence of evidence is in fact evidence of absence. If observing some evidence tomorrow would make your hypothesis more likely, then if tomorrow comes and you do not observe the evidence, your hypothesis becomes less likely. So if you can recognize the circumstances in which your hypothesis becomes more likely, you should be able to recognize the circumstances in which it becomes less likely.

For example, in the previous post, most people would recognize that (2) is evidence against (1), but recognizing this appears to be more difficult for Mormons. Nonetheless, if it had turned out that the Book of Abraham was in fact an accurate translation of an ancient Egyptian manuscript, this would have been evidence favoring Mormonism, and there can be no doubt that Mormons would have recognized it as such. Consequently, if they can recognize that this would have favored their position, they should be able to recognize that the actual fact (2) is evidence against it.

You cannot have it both ways. If you concede that getting what you ask for every time you pray to your guardian angel would be evidence for his existence, then not getting what you ask for is evidence against his existence. Of course such negative evidence is not necessarily very strong, and this may in fact be the point of the linked post.

The Evidence Does Not Change Sides

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Suppose someone holds the following position:

(1) Mormonism is the true religion.

Then he discovers this fact:

(2) Joseph Smith claimed to have translated the Book of Abraham from an Egyptian manuscript, but in reality the Egyptian manuscript was about something completely different.

Now this doesn’t look good. In fact, it looks exactly as though the Book of Abraham is a complete fraud, which seems to imply that position (1) is false. In other words, (2) is strong evidence against (1).

So our protagonist modifies his position like this:

3) Mormonism is the true religion, and Smith interpreted the Egyptian manuscript by divine inspiration, revealing its spiritual sense.

Now he notices something. After the discovery of fact (2), position (3) becomes more probable than it originally was, since part of position (3) is now verified to be definitely true, namely the fact that the book of Abraham is a not a literal translation of the Egyptian manuscript. Thus, the original disturbing fact which seemed to be evidence against his position, is now evidence in favor of his new position! And the new position includes position (1), so there is no need to change a thing!

This reasoning is technically valid, of course, but our protagonist is overlooking a few things.

First of all, (1) is in itself more probable than (3), since (3) requires the truth of (1) and something else in addition.

Second, after the discovery of (2), (1) becomes less probable, likely significantly less probable, than it was before. This fact remains unchanged by the rest of the process.

Third, (3) does indeed become more probable than it originally was, after the discovery of (2). However, (3) was less probable than (1) in the first place, and even after it becomes more probable, it remains less probable than (1) originally was, and it also remains less probable than (1) became after the discovery of (2). This is necessary because (3) is nothing but a particular way that (1) could be true. So by adopting the new position, our protagonist has not benefited by (2) in the way that he believes. Rather, he ends up holding a position that is even less probable than claiming that Mormonism is true, admitting that the Book of Abraham is not a valid translation of the Egyptian manuscript, admitting that this makes his original position regarding Mormonism less probable, and making no other changes.

In other words, the evidence does not change sides.

The Evidence is Not Automatically on Your Side

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One thing is evidence for a second thing if the second thing is more probable given the first, than the second without considering the first. Thus the fact that you are reading this blog post is evidence that you are a native English speaker, since someone reading the post is more likely to be a native English speaker than a random person is.

One common mistake is to think that there cannot be evidence for something false; but my position is true and the opposite is false; therefore there cannot be any evidence against my position. Thus people say things like, “Evolutionary belief is a remarkable and largely unexplained phenomenon. It is a belief held by most intellectuals all over the world, despite the fact that there is no real scientific evidence for it at all.” Again, someone holding another position says, “Critics of evolution claim that it is just a theory for which there is no proof. It is true there is no definitive proof, and nor is there likely to be, but there is a vast amount of evidence in its favour. Whether you choose to believe it is sufficient is up to you, but it is there. By contrast, there is no scientific evidence for creationism.”

The claim that your opponent’s position has no evidence for it is always false, without exception. For the very fact that your opponent holds the position is evidence for it, since a position that someone holds is more likely to be true than a random position that no one holds. But even apart from this, given any particular position that real people hold, we can expect to be able to find any number of facts that make more it more likely than it would be without those facts, even if the thing is absolutely false. Thus if you buy a lottery ticket, it is evidence that you will win the lottery, since it becomes more likely that you will win, having a ticket, than not having one. But ordinarily you won’t win anyway, despite your evidence for it.