Comment on Welcome to Less Wrong! (2012)

Tue, 29 May 2012 13:13:58 GMT

Thank you for the chronological suggestion; I think that will help it to make sense.

I would comment that the consistency theory for posteriors only guarantees convergence to "the truth" WHEN the likelihood terms are capable of making distinctions between the different possible models. If data is unavailable that distinguishes them... no convergence, just stabilization to potentially different subjective posteriors supported on the indistinguishable set. Since all we care about is making decisions, this is usually a distinction with no difference, except if there is a future event to come where they will make different predictions.

Accordingly my "God hypothesis" posterior probability has not updated in a very long time. I am, for example, not aware of what evidence there exists AT ALL with which to update this prior, but I suppose this is just because my posterior mass, conditional on the God hypothesis, came to rest, long ago, on a very non-falsifiable, (arguably science-compatible) version of belief in God. I.e. belief in a God who "operates all in accord with His will", but who also does this exactly through the naturalistic mechanisms that we see operating all the time and not through miraculous interventions. Or, to be more precise, another indistinguishable possibility is that God is perpetrating miraculous interventions continuously, and just consistently wills to do them in exactly the way that the true but unknown "laws of nature" predict. It remains possible that God could revoke this policy at some future time: this does not require much Kolmogorov complexity. I can rule out the ordinary miraculous part of the mass ("why did it happen? Because God wanted it that special way in this special case") because it's not predictive. Accordingly my posterior does not give me "the mathematical right" to assert God's non-existence.

Indeed, unless posterior mass is actually 0, I don't think this is a matter of "mathematical rights" anyway. We're making decisions under uncertainty, and we have some model for what the loss is if we make a mistake. You have, for example, the right to, say, not believe in God but not want to talk about it anymore, and otherwise act in accord with your belief, but I don't see how it gives you the right to assert God's non-existence. Sometimes we assign mass 0 to some possibilities, effectively, because we are not perfectly rational Bayesians, either we'd never thought of those possibilities, or it was computationally expedient to ignore them. In either of these cases we have no such "right".

If I had the misfortune to have my prior "initialized" by my environment somewhere with a "vengeful God" hypothesis, who waits in Secret to test if you are really "one of the chosen" faithful, I'd be in a more difficult predicament. Hopefully, if I'd been born in such a sect, they'd have had some other clearly falsifiable beliefs that come along with the dogmatic ride, so that I'd have a way to identify the error. The worrisome part here, I notice, is that my thinking is rather driven by how it was initialized.

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