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Thinking and Deciding 5: Normative Theory and Probability

09/02/2013

Probability here is “a numerical measure of the strength of belief in a certain proposition” (104).

What is probability?

Two problems need to be solved to answer this question. 1) How do we construct well-justified probability statements? 2) How do we evaluate probability judgments after discovering the truth?

There are three competing theories that answer these questions in different ways. Frequency theory claims that probability statements are based on the relative frequency of events. If a die roll comes up 3 1/6 of the time, then it has a 1/6 probability. We evaluate our probability judgments by comparing them to the frequency of events.

The first problem is that we don’t always know what reference class to use. Should I only compare the die rolls of this one particular die? Should I only compare them when they are rolled exactly the same way? Different probabilities arise depending on what class the roll falls into. Second, flipping a coin and getting 7 heads and 3 tails would warrant a probability of .7 for heads on the next flip. This goes against (according to Baron) strong intuitions. I’d just say that it fails predictions.

One way to try to fix this is to say that probability is based on the frequency taken by an infinite number of these events. But how do we judge this to be true on frequentism if we don’t actually perform this test? It starts to change into personal theories quickly.

The logical theory splits possibility space into spaces of equal likelihood. So the results of three coins flipped have 9 possible outcomes, each equally likely. Here, likelihood is an objective fact about the universe.

The problem with this view is that only in very limited situations can we use it to make judgments. How likely rain is tomorrow, or how likely my dog is to have fleas if I go to the park are nearly impossible to split into equally likely possibilities.

The personal theory sees probability as a reflection of the personal judgment of likelihood of a situation. That means different people with different sets of knowledge and beliefs may reasonably make different probability judgments. It makes it difficult to evaluate probability judgments on this view, but Baron hints at Bayes’ theorem as a way of solving this problem.

This view allows any and every event to be assigned a probability. When there is no way to weigh one possibility as more likely than another, then we assign it equal likelihood by the principle of insufficient reason.

Well-justified probability judgments

One way to count probability judgments as well-justified is by making sure they cohere with each other. One cannot assign more than 1 probability to an exhaustive list of probabilities. Raising one lowers one or more of the rest. Lowering one raises one or more of the rest. People are very bad at doing this. Other logical rules (Bayes theorem) apply.

Evaluating probability judgments

Calibration serves as one method of evaluation. Well calibrated judgments are in line with what is observed. 75% confidence in rain should predict rain 75% of the time.

Bayes’ Theorem

Baron sketches out Bayes’ theorem with a typical base rate fallacy example of cancer patient testing. He also goes on to answer how frequentist probability can matter for personal probability theory holders. Basically in line with Bayes theorem, one adjusts levels of confidence as more information comes in. I find it hard to see how this was a problem in the first place.

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