MinerGlitch

Ok, got it. Let's start by assuming the researcher knows the answer, but must test to prove it. For each hypothesis, there is a minimum cost set of tests which proves the hypothesis. Lets call this the Cost Under Perfect Information (CUPI) of the hypothesis. Now, let’s develop a naïve strategy for a less knowledgeable researcher. For each hypothesis, this researcher calculates the CUPI divided by the ex-ante probability of the hypothesis. The tests with the lowest relative costs are done first, then the ones with the next lowest relative cost, etc, until a hypothesis is proved. Under this naïve approach, there are at least as many tests performed for each hypothesis as there are under CUPI. Let’s call the cost of these tests the Cost Under No Information (CUNI) of the hypothesis. Next, calculate the value of perfect information (VPI) for each hypothesis, which is simply CUNI – CUPI. Weight the VPI by the probability of each hypothesis, to give the expected value of perfect information (EVPI). Similarly, calculate the Expected Cost Under Perfect Information (ECUPI). Determine for all tests by how much the EVPI is expected to be decreased by performing the test. This reduction is the value of the test. Also calculate the net cost of the test. This will be the cost, less, if it's a test in a minimum cost set, the expected reduction in the ECUPI. Divide the value of each test by its net cost, yielding the efficiency of the test. The optimal strategy is then simply to do the most efficient test first.

A particular problem must be diagnosed. There are different hypotheses h1,h2,....hj as possible solutions, with required probabilities for acceptance p1, p2, ..., pj. The hypotheses are mutually exclusive and exhaustive, that is P(h1) + P(h2) + ... + P(hj) = 1 at all times. There are different tests that can be performed, t1,t2,....,tk to provide information to assist in the diagnosis. The tests cost c1,c2,....,ck. The result of each test r1,r2,...,rk is qualitative taking on only three possible values: true, false, or not yet performed. The probability vector that a given hypothesis is correct <P(h1),P(h2),..,P(hj)> is a known function f(r1,r2,...,rk). Tests are performed one after the other until a hypothesis hn is proven at the required level, i.e fn(r1,r2,...,rk) > pn. If all tests are performed, there exists at least one <r1,r2,...,rk> that would provide for acceptance of each hypothesis. What is the strategy for selecting the next test to be performed in each case which minimises the expected total cost of diagnosis?