Here's the way I understand it: 1. The model says there exists some true value of a parameter, say, how much more likely you are to get cancer if you use hair

Here's the way I understand it: 1. The model says there exists some true value of a parameter, say, how much more likely you are to get cancer if you use hair dye. 2. You run an experiment with two groups, hair dye, and no hair dye. 3. If you were to run this experiment 100 times, you would expect that 5 of those times, the true value wouldn't be within the confidence interval, and 95 times, it would be. This type of analysis doesn't tell what you really want to know: how likely is it that hair dye causes cancer. What you do know is that you should expect that if you run a bunch of experiments, you should expect 5% of your 95% confidence intervals to be "wrong." To have a degree of belief in something, you need a prior. If I run a hazard ratio analysis for skydiving without a parachute, and the 95% CI includes a hazard ratio of 1.0, I'm still not going to believe that this is a safe activity. If I run it on drinking a glass of water, and the 95% CI doesn't include 1.0, I'm still going to believe that drinking a glass of water is safe.