I'm Starting To Get It

After considerably more reading, this is the way I'm understanding it. Let me know what I'm missing or getting wrong.

The key to Bayes’ theorem is that it makes explicit the relationship between two events, and the probability of one event given that the other event has occurred.

Here’s a simple example: Say you want to know if it rained last night, so you go outside and touch the grass to see if it’s wet. Before you touch the grass there is some probability that it rained during the night, maybe based on a forecast of 40% chance of rain. But after you touch the grass and feel that it is wet, there is a new (higher) probability that it rained. Bayes’ theorem gives you a way to calculate the new probability that it rained last night, given the evidence of wet grass.

You can’t conclude that it rained last night based on the fact that the grass is wet, because the sprinklers may have come on or it may just be dew. But if you know something about the relationship between the event of the grass being wet and the event of it having rained, then you can calculate how likely it is that rain is the cause of the wet grass (CORRECTION - that reference to 'cause' is objectionable, we're only discussing correlation). The real insight of Bayes’ theorem is that the likelihood that rain is the cause of the wet grass is related to the probability of the grass being wet when it hasn’t rained.

If you happen to know that there is a very low probability of the grass being wet in the morning after a night with no rain (maybe you don’t have sprinklers, and you live in a dry climate with very little dew), then wet grass is a strong indicator of rain. But if there is a high probability of the grass being wet on a morning after no rain, then wet grass is a very weak indicator of rain.

This is fairly intuitive, and so maybe it doesn’t seem very revolutionary. However, what Bayes’ theorem does is it makes this kind of reasoning explicit and calculable. Bayes’ theorem justifies this kind of reasoning by formally spelling out how and why it works. An understanding of Bayes’ theorem also helps one avoid mistakes of probabilistic reasoning, e.g. thinking that wet grass is a stronger indicator of rain than it really is.