By Alexander J. W.

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**Example text**

The result can be near 1/2, representing the state of complete uncertainty where originally that probability is near 0 (or 1), a state of near certainty about Y . We demonstrate this in the following example. 1. 3 Uncertainty Versus Information 31 Suppose an individual assesses her uncertainty about X as represented by a uniform probability on [0,1], denoted X ∼ U [0, 1]. In particular, this individual would be quite certain (90% sure) that X does not lie in A = [0, 2/20]. However, that individual learns that X lies in B = [1/20, 2/20].

But how should it be quantiﬁed? ” Within statistics, the Bayesian paradigm, embraced by the above quotations of Lindley as well as O’Hagan and developed below, seems ideal for discussing uncertainty (and more generally for risk assessment). There, roughly speaking, uncertainty is equivalent to randomness and the degree of uncertainty about any aspect of the world, past, present, and future, can be expressed through a probability distribution. That is the paradigm on which this book is based. Therefore our discussion focuses on an uncertain, and hence random object of interest, Y , that could be a matrix, vector, or even real number.

These parameters, unlike say Σ above, are found not in the distribution that describes the distribution of the sample values directly, but rather they are parameters in the prior structure. The latter provides a distribution on the ﬁrst-level parameters like Σ and express LSZ’s uncertainty about them. (See Chapter 3. ) It turns out these parameters can be estimated from the data. To do so, they used a standard method called the EM algorithm. , a distribution for the unmeasured responses of interest.