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R. Kass/S12 P416 Lecture 5 1 l Suppose we are trying to measure the true value of some quantity (x T ). u We make repeated measurements of this quantity {x 1, x 2, … x n }. u The standard way to estimate x T from our measurements is to calculate the mean value: and set x T = x. DOES THIS PROCEDURE MAKE SENSE??? The maximum likelihood method (MLM) answers this question and provides a general method for estimating parameters of interest from data. l Statement of the Maximum Likelihood Method u Assume we have made N measurements of x {x 1, x 2, … x n }. u Assume we know the probability distribution function that describes x: f(x, ). u Assume we want to determine the parameter. (e.g. we want to determine the mean of a gaussian) MLM: pick to maximize the probability of getting the measurements (the x i 's) that we got! l How do we use the MLM? u The probability of measuring x 1 is f(x 1, )dx u The probability of measuring x 2 is f(x 2, )dx u The probability of measuring x n is f(x n, )dx u If the measurements are independent, the probability of getting the measurements we got is: The Maximum Likelihood Method (Taylor: Principle of maximum likelihood) We can drop the dx n term as it is only a proportionality constant.

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R. Kass/S12 P416 Lecture 5 2 u We want to pick the that maximizes L: u Often easier to maximize lnL. Both L and lnL are maximum at the same location. we maximize lnL rather than L itself because lnL converts the product into a summation. The new maximization condition is: n could be an array of parameters (e.g. slope and intercept) or just a single variable. n equations to determine range from simple linear equations to coupled non-linear equations. l Example: u Let f(x, ) be given by a Gaussian distribution function. u Let = be the mean of the Gaussian. We want to use our data+MLM to find the mean,. u We want the best estimate of from our set of n measurements {x 1, x 2, … x n }. u Lets assume that is the same for each measurement. u The likelihood function for this problem is: gaussian pdf

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R. Kass/S12 P416 Lecture 5 3 We want to find the that maximizes the log likelihood function: u If are different for each data point then is just the weighted average: Average! The MLM method says that calculating the average is the best thing we can do in this situation. factor out since it is a constant Weighted Average dont forget the factor of n

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R. Kass/S12 P416 Lecture 5 4 l Example: Poisson Distribution u Let f(x, ) be given by a Poisson distribution. u Let = be the mean of the Poisson. u We want the best estimate of from our set of n measurements {x 1, x 2, … x n }. u The likelihood function for this problem is: u Find that maximizes the log likelihood function: Some general properties of the Maximum Likelihood Method For large data samples (large n) the likelihood function, L, approaches a Gaussian distribution. Maximum likelihood estimates are usually consistent. For large n the estimates converge to the true value of the parameters we wish to determine. Maximum likelihood estimates are usually unbiased. For all sample sizes the parameter of interest is calculated correctly. Maximum likelihood estimate is efficient: the estimate has the smallest variance. Maximum likelihood estimate is sufficient: it uses all the information in the observations (the x i s). The solution from MLM is unique. Bad news: we must know the correct probability distribution for the problem at hand! Average!

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R. Kass/S12 P416 Lecture 5 5 Maximum Likelihood Fit of Data to a Function (essentially Taylor Chapter 8) l Suppose we have a set of n measurements: u Assume each measurement error ( ) is a standard deviation from a Gaussian pdf. u Assume that for each measured value y, theres an x which is known exactly. u Suppose we know the functional relationship between the ys and the xs: n,...are parameters that we are trying determine from our data. MLM gives us a method to determine,... from our data. l Example: Fitting data points to a straight line. We want to determine the slope ( ) and intercept ( ). two linear equations with two unknowns u Find and by maximizing the likelihood function L:

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R. Kass/S12 P416 Lecture 5 6 For the sake of simplicity assume that all s are the same ( 1 = 2 = i ): We now have two equations that are linear in the unknowns, : in matrix form These equations correspond to Taylors Eq

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R. Kass/S12 P416 Lecture 5 7 l EXAMPLE: A trolley moves along a track at constant speed. Suppose the following measurements of the time vs. distance were made. From the data find the best value for the speed (v) of the trolley. u Our model of the motion of the trolley tells us that: Time t (seconds) Distance d (mm) u We want to find v, the slope ( ) of the straight line describing the motion of the trolley. u We need to evaluate the sums listed in the above formula: d 0 = 0.8 mm best estimate of the speed best estimate of the starting point The variables in this example are time (t) and distance (d) instead of x and y.

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R. Kass/S12 P416 Lecture 5 8 The line in the above plot best represents our data. Not all the data points are "on" this line. The line minimizes the sum of squares of the deviations ( ) between a line and our data (d i ): i =data-prediction= d i -(d 0 +vt i ) minimize: Σ[ i 2 ] same as MLM! Often we call this technique Least Squares (LSQ). LSQ is more general than MLM but not always justified. The MLM fit to the data for d=d 0 +vt

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