Modeling of Data. Basic Bayes theorem Bayes theorem relates the conditional probabilities of two events A, and B: A might be a hypothesis and B might.

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Presentation transcript:

Modeling of Data

Basic Bayes theorem Bayes theorem relates the conditional probabilities of two events A, and B: A might be a hypothesis and B might be some data. so that P(A|B) expresses the probability of a hypothesis, given the data.

Least Squares as a Maximum Likelihood Estimator Suppose we have N points of data M adjustable parameters The goal is to fit data points to the model.

Least Squares as a Maximum Likelihood Estimator The model predicts a functional relationship between the measured independent and dependent variables. We want to find values for parameters, such that: is minimized.

Chi Square fitting suppose that each data point y i has a measurement error that is independently random and distributed as a normal (Gaussian) distribution around the true model y(x) And suppose that the standard deviations of these normal distributions are the same for all points. Then the probability of the data set is the product of the probabilities of each point. …..(1)

Chi Square fitting The most probable model, then, is the one that maximizes equation (1) or, equivalently, minimizes the negative of its logarithm,

Chi Square fitting Since N,, and are all constants, minimizing this equation is equivalent to minimizing

Fitting Data to a Straight Line fitting a set of N data points to a straight-line model using the chi-square Assume that the uncertainty associated with each measurement yi is known

Fitting Data to a Straight Line Taking derivatives of with respect to a; b vanish:

Fitting Data to a Straight Line Define these sums: to simplify the questions to :

Fitting Data to a Straight Line The solution of these two equations in two unknowns is calculated as:

Estimate the probable uncertainties Each data point contribute to a bit of uncertainty to the parameters. the variance in the value of any function is

Estimate the probable uncertainties For the straight line, the derivatives of a and b with respect to y i can be directly evaluated from the solution: Summing over the points: