Nonlinear MAP

The algorithms supplied in the package Statistics'NonlinearFit" only perform parameter estimation based on least-squares minimization. On the other hand, the function FindMinimum uses a variety of methods, including the LevenbergMarquardt method, to determine the local minima of a user defined function. So, if we define our own MAP function we can use FindMinimum for parameter estimation, as follows:

FindMinimum[ Searches for a local minimum f, {x, Xo}, {y, yo}] in a function of several variables

Minimizing merit functions.

Q: Is it possible to fit the Michaelis-Menten equation onto enzyme kinetic data assuming that we have prior knowledge about the Vmax of the enzyme? Suppose that the prior knowledge is that the errors in Vmax are normally distributed with a mean of 10 mmol L-1 min-1 and a variance that is 10 times less than the variance of the experimental data. The synthetic data from the previous example (dataEnzNoisy) fulfills these criteria, since they were constructed with these properties in the first place.

A: We have prior knowledge about the value of Vmax so we use MAP estimation rather than least-squares estimation. First, define the MAP merit function, recalling that a MAP function for normally distributed measurement errors and normally distributed prior parameter probabilities is given by Eqn [6.14].

Hence for the data in dataEnzNoisy the MAP merit function is given by Eqn [6.14], as follows:

Vmax a conc = Transpose [dataEnzNoisy] ; velMeas = Transpose[dataEnzNoisy] pj ; velPred = Map[vo , conc]; MAP [Km_, Vmax_] : =

Now apply the FindMinimum function to estimate the Michaelis-Menten parameters. FindMinimum[MAP[Km, Vmax] , {Vmax, 10.}, {Km, 3.}]

Notice how the estimate of Vmax is much closer to 10 than in the previous example. The fact that the variance of our prior distribution for Vmax was low compared to the experimental data means that deviations from this prior belief were heavily weighted in the MAP merit function.

The important topic of estimating the standard errors associated with parameter estimates that are obtained by using MAP are discussed in Section 6.8.

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