Let T1_21 and T2_21 be the two theories’ predictions for the twenty-first experiment.
As you note, if all else is equal, our prior beliefs about P(T1_21) and P(T2_21) -- the odds we would’ve accepted on bets before hearing T1s and T2′s predictions—are relevant to the probability we should assign after hearing T1′s and T2′s predictions. It takes more evidence to justify a high-precision or otherwise low-prior-probability prediction. (Of course, by the same token, high precision and otherwise low-prior predictions are often more useful.)
The precision (or more exactly, the prior probability) of the predictions T1 and T2 assign to the first twenty experimental results are also relevant. The precision of these tested predictions, however, pulls in the opposite direction: if theory T1 made extremely precise, low-prior-probability predictions and got them right , this should more strongly increase our prior probability that T1′s set of predictions is entirely true. You can formalize this with Bayes’ theorem. [However, the obvious formalization only shows how probability of the conjunction of all of T1′s predictions increases; you need a model of how T1 and T2 were generated to know how indicative each theory’s track record is of its future predictive accuracy, or how much your beliefs about P(T1_21) specifically should increase. If you replace the scientists with random coin-flip machines, and your prior probability for each event is (1/2), T1′s past success shouldn’t increase your P(T1_21) belief at all.]
As to whether there is a single “best” metric for evaluating theories, you are right that for any one expert, with one set of starting (prior) beliefs about the world and one set of data with which to update those beliefs, there will be exactly one best (Bayes’-score-maximizing) probability to assign to events T1_21 and T2_21. However, if the two experts are working from non-identical background information (e.g., if one has background knowledge the other lacks), there is no reason to suppose the two experts’ probabilities will match even if both are perfect Bayesians. If you want to stick with the Solomonoff formalism, we can make the same point there: a given Solomonoff inducer will indeed have exactly one best (probabilistic) prediction for the next experiment. However, two different Solomonoff inducers, working from two different UTM’s and associated priors (or updating to two different sets of observations) may disagree. There is no known way to construct a perfectly canonical notion of “simplicity”, “prior probability” or “best” in your sense.
If you want to respond but are afraid of the “recent comments” limit, perhaps email me? We’re both friends of Jennifer Mueller’s (I think. I’m assuming you’re the Tyrrell McAllister she knows?), so between that and our Overcoming Bias intersection I’ve been meaning to try talking to you sometime. annasalamon at gmail dot com.
Hi Tyrrell,
Let T1_21 and T2_21 be the two theories’ predictions for the twenty-first experiment.
As you note, if all else is equal, our prior beliefs about P(T1_21) and P(T2_21) -- the odds we would’ve accepted on bets before hearing T1s and T2′s predictions—are relevant to the probability we should assign after hearing T1′s and T2′s predictions. It takes more evidence to justify a high-precision or otherwise low-prior-probability prediction. (Of course, by the same token, high precision and otherwise low-prior predictions are often more useful.)
The precision (or more exactly, the prior probability) of the predictions T1 and T2 assign to the first twenty experimental results are also relevant. The precision of these tested predictions, however, pulls in the opposite direction: if theory T1 made extremely precise, low-prior-probability predictions and got them right , this should more strongly increase our prior probability that T1′s set of predictions is entirely true. You can formalize this with Bayes’ theorem. [However, the obvious formalization only shows how probability of the conjunction of all of T1′s predictions increases; you need a model of how T1 and T2 were generated to know how indicative each theory’s track record is of its future predictive accuracy, or how much your beliefs about P(T1_21) specifically should increase. If you replace the scientists with random coin-flip machines, and your prior probability for each event is (1/2), T1′s past success shouldn’t increase your P(T1_21) belief at all.]
As to whether there is a single “best” metric for evaluating theories, you are right that for any one expert, with one set of starting (prior) beliefs about the world and one set of data with which to update those beliefs, there will be exactly one best (Bayes’-score-maximizing) probability to assign to events T1_21 and T2_21. However, if the two experts are working from non-identical background information (e.g., if one has background knowledge the other lacks), there is no reason to suppose the two experts’ probabilities will match even if both are perfect Bayesians. If you want to stick with the Solomonoff formalism, we can make the same point there: a given Solomonoff inducer will indeed have exactly one best (probabilistic) prediction for the next experiment. However, two different Solomonoff inducers, working from two different UTM’s and associated priors (or updating to two different sets of observations) may disagree. There is no known way to construct a perfectly canonical notion of “simplicity”, “prior probability” or “best” in your sense.
If you want to respond but are afraid of the “recent comments” limit, perhaps email me? We’re both friends of Jennifer Mueller’s (I think. I’m assuming you’re the Tyrrell McAllister she knows?), so between that and our Overcoming Bias intersection I’ve been meaning to try talking to you sometime. annasalamon at gmail dot com.
Also, have you read A Technical Explanation ? It’s brilliant on many of these points.