Prior Information into the Likelihood

Science is rarely done in a vacuum. When an experiment is conducted, the scientist often has some notion about what the outcome will be. In modeling, the analyst may have some idea of what the model parameters will be. For example, at the very least, one can make the assumption that the rate constants in a compartmental model are all positive—that the rate constants cannot be negative. The section on Constrained Optimization in this chapter illustrated how such constraints can be incorporated into the fitting process. However, sometimes the analyst has even more knowledge about the value of a model parameter, such as the distribution of the parameter. Such a distribution is referred to as the prior distribution. For example, suppose in a previous study clearance was estimated to be normally distributed with a mean of 45 L/h and a standard deviation of 5 L/h. 

Bayesian statistics has at its heart the following fundamental equality 

In other words, the product of the likelihood of the observed data [e.g., Eq. (3.12)] and the prior distribution is proportional to the probability of the posterior distribution, or the distribution of the model parameters taking into account the observed data and prior knowledge. Suppose the prior distribution is p-dimensional multivariate normal with mean 0 and variance O, then the prior can be written as 

Recall that the likelihood function for a model with Gaussian random errors (the observed data) can be written as 

Minimizing Eq. (3.130) with respect to 0 then produces an estimate of 0 that incorporates prior knowledge about 0 into the estimate combined with the observed data likelihood, thereby reflecting all the information that is relevant to 0. SAAM II forces the user to assume a normal prior distribution, while Adapt II allows for normal and log-normal prior distributions. But, if access can be made to general optimization software, such as within MATLAB, then an analyst is not limited to specific prior distributions, or indeed, normal likelihood functions. The reader is referred to Lindsey (2000 2001) for such examples. 

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