Appendix A: A Bayesian Investigation Loop

In this appendix we give a heuristic algorithm that integrates Bayesian updating into a small model of an investigation.

Here is how it works:

Suppose we begin with a hypothesis space:

$$ H_1,\ldots,H_n. $$

Each hypothesis has a prior weight.

When new evidence $E$ becomes available, we ask:

$$ P(E\mid H_i) $$

for each hypothesis.

Bayes' theorem then constrains how the weights change:

$$ P(H_i\mid E) =\frac{P(E\mid H_i)P(H_i)} {P(E)}. $$

The posterior can then become the prior for the next update.

A simplified investigation might look like this:

1. Begin with a represented set of possibilities.
2. Assign or accept initial weights.
3. Observe or represent new evidence.
4. Ask how likely that evidence would be under each represented possibility.
5. Update the weights.
6. Continue with further evidence.

As long as the hypothesis space remains fixed, Bayesian updating constrains the movement of the weights.

But the investigation can also change in another important way.

A new possibility may become available.

Suppose we began with:

$$ H_1,H_2,H_3 $$

and later realize that:

$$ H_4 $$

also needs to be considered.

The new possibility has to be represented, some initial weight has to be given to it, the weights of the existing hypotheses have to be reconsidered relative to the enlarged space, and relevant likelihoods have to be supplied.

So a more realistic heuristic is:

Start with represented possibilities
            |
            v
       New evidence
            |
            v
How likely is the evidence under each hypothesis?
            |
            v
      Calculate probability of evidence
            |
            v
      Bayesian update
            |
            v
Are the represented possibilities still adequate?
         /       \
       yes        no
        |          |
        |          v
        |    revise possibility space
        |    introduce new possibilities
        |    reconsider weights
        |    supply new likelihoods
        |          |
        +----------+
            |
            v
      continue inquiry

The two routes in this heuristic are different.

As long as the represented possibilities remain in place, Bayesian updating constrains what happens to their weights.

The other route is less constrained.

A new possibility may arise from observation, experience, analogy, another person's suggestion, a failed prediction, another model, or something else.

Once it enters the investigation, the probability model has to be rebuilt around the enlarged possibility space.