What's The Weight Of Evidence?

Why do people use "weight of evidence"?

• The weight of evidence is defined [math]\mathit{woe}(h:e)=\ln \frac{p(e \mid h)}{p(e \mid \overline h)}[/math] What's the difference between PMI?

• Answer:

  In your woe, you have in hand the evidence, based on your observations, and you are trying to infer from that whether h is more likely to be true or the complement of h is more likely to be true (i.e., h is false). In order to make that determination, you go through a bit of a mental inversion by declaring that h is more likely to be true than its complement if, when h is true, the probability of seeing the evidence that you saw is higher than the probability of seeing that evidence in the opposite case when h is not true. As an example, if you see someone at the end of a Niners-Packers football game cheering and hooting after, say, a victory of the Niners, and you are trying to guess whether this guy is a Niners fan or a Packers fan (and he's not wearing any obvious jersey or face paint, etc.), then you'd ask yourself: - If he's a Niners fan and the Niners just won, what's the probability that I'd see him cheering and hooting the way I am seeing this guy? You reason with yourself that roughly half the Niners fans are extraverts who cheer and hoot upon victory and the other half are introverts who are happy too but don't demonstrate it as overtly. So the numerator of your woe above is 0.5 or so. - If he's a Packers fan and the Niners just won, what's the probability that I'd see this guy cheering and hooting as I am observing now? You reason that perhaps half the Packers fans are extraverts and are hissing and booing right now, and the other half or so are introverts who take defeat in strides with stoicism. But it would be a very rare bird indeed who is a declared Packers fan and yet shows explicit signs of joys upon defeat, so you assign some odd-bird probability to this observation, say 0.001, and that's now the denominator of your ratio above. Your woe is now ln (0.5/0.001), which is over 6, a very big number for a Log. So you conclude, based on the evidence (behavior of the fan), that he is much more likely to be a Niners fan (i.e., h is true) than a Packers fan (h-bar is true). As to pmi, it is something else altogether: it measures the plausibility that 2 variables are independent (or how far from that they are), by taking the log of the ratio of their joint probability (as measured) to their probability if they'd been independent (which would be the product of their individual marginal probabilities). So other than both having similar mathematical forms (logs of probability ratios), the underlying use cases are typically distinct.