Expected Value of Sample Information - Formulation

Formulation

Let

\begin{array}{ll} d\in D & \mbox{the decision being made, chosen from space } D \\ x\in X & \mbox{an uncertain state, with true value in space } X \\ z \in Z & \mbox{an observed sample composed of } n \mbox{ observations } \langle z_1,z_2,..,z_n \rangle \\ U(d,x) & \mbox{the utility of selecting decision } d \mbox{ from } x \\ p(x) & \mbox{your prior subjective probability distribution (density function) on } x \\ p(z|x) & \mbox{the conditional prior probability of observing the sample } z \end{array}

It is common (but not essential) in EVSI scenarios for, and, which is to say that each observation is an unbiased sensor reading of the underlying state, with each sensor reading being independent and identically distributed.

The utility from the optimal decision based only on your prior, without making any further observations, is given by

E = \max_{d\in D} ~ \int_X U(d,x) p(x) ~ dx

If you could gain access to a single sample, the optimal posterior utility would be:

E = \max_{d\in D} ~ \int_X U(d,x) p(x|z) ~ dx

where is obtained from Bayes' rule:

p(x|z) = {{p(z|x) p(x)}\over{p(z)}}
p(z) = \int p(z|x) p(x) ~ dx

Since you don't know what sample would actually be obtained if you were to obtain a sample, you must average over all possible samples to obtain the expected utility given a sample:

E = \int_Z E p(z) dz = \int_Z \max_{d\in D} ~ \int_X U(d,x) p(z|x) p(x) ~ dx ~ dz

The expected value of sample information is then defined as:

\begin{array}{rl} EVSI & = E - E \\ & = \left(\int_Z \max_{d\in D} ~ \int_X U(d,x) p(z|x) p(x) ~ dx ~ dz\right) - \left(\max_{d\in D} ~ \int_X U(d,x) p(x) ~ dx\right) \end{array}

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