The Goal of Bayesian Inference: Quantify and manipulate your degrees of beliefs. In other words, Bayesian inference is the Analysis of Beliefs.
Bayesian inference is no more about beliefs than logic is (or any scientific inference, really). "M(G) AND C(M) => C(G)" can be rendered as "If you believe that glass is a metal, and you believe that metals are good conductors, then you should also believe that glass is a good conductor". Scientists omit the "if you believe" out of conciseness.
Some subjective Bayesians will tell you that their job is to produce the above. Then they're done. "You said you believe that glass is a metal, so I put that into my Bayesian inference procedure, and it says that you should also believe that glass is a good conductor."
But this is not what science is about! Obviously, "glass is a conductor" strongly contradicts empirical data. We have to challenge every assumption, and possibly change models!
This is why smart Bayesians check the fit of their model, and I would strongly recommend Gelman's Induction and Deduction in Bayesian Data Analysis to any statistician interested in that perspective. It places Bayesianism squarely in the paradigm of traditional scientific analysis.
"Metal" only refer to a set of kinds of matter. A set we shaped because it helps us make useful inferences without using too much brain power. Like the fast rules: "Most metals are good conductors", "Most metals are strong", "Most metals are hard", "Most metals are heavy".
Then someone comes and shows you that new material called "glass" that is heavy, hard, and strong (this one is bullet proof). You'd be quick to infer that it is a metal, and therefore probably a good conductor. But didn't we tell you that most metals are opaque?
Bayesian inference is no more about beliefs than logic is (or any scientific inference, really). "M(G) AND C(M) => C(G)" can be rendered as "If you believe that glass is a metal, and you believe that metals are good conductors, then you should also believe that glass is a good conductor". Scientists omit the "if you believe" out of conciseness.
Some subjective Bayesians will tell you that their job is to produce the above. Then they're done. "You said you believe that glass is a metal, so I put that into my Bayesian inference procedure, and it says that you should also believe that glass is a good conductor."
But this is not what science is about! Obviously, "glass is a conductor" strongly contradicts empirical data. We have to challenge every assumption, and possibly change models!
This is why smart Bayesians check the fit of their model, and I would strongly recommend Gelman's Induction and Deduction in Bayesian Data Analysis to any statistician interested in that perspective. It places Bayesianism squarely in the paradigm of traditional scientific analysis.
http://www.rmm-journal.de/downloads/Article_Gelman.pdf