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2.2 Three Goals for Inference from Data 13 2.2.1 Estimation of Parameter Values One goal we may have is deciding the extent to which we should believe in each of the possible values of an underlying parameter. In the case of the coin, we may use the observed data to determine the extent to which we should believe that the bias is 20%, 50%, or 80%. What we are determining is how much we believe in each of the available parameter values. In most real appli- cations, we allow for a continuum of possible biases from zero to one, and the Bayesian mathematics reveal the credibility of every possible value on that continuum. Because the flip of the coin is a random process, we cannot be certain of the underlying true probability of getting heads. So our posterior beliefs are an esti- mate. The posterior beliefs typically increase the magnitude of belief in some parameter values, while lessening the degree of belief in other parameter val- ues. So this process of shifting our beliefs in various parameter values is called estimation of parameter values. 2.2.2 Prediction of Data Values