R. Kass/W03 P416 Lecture 5 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 . MLM: pick  to maximize the probability of getting the measurements (the x i 's) that we did! 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 did is: Lecture 5 The Maximum Likelihood Method We can drop the dx n term as it is only a proportionality constant

R. Kass/W03 P416 Lecture 5 u We want to pick the  that maximizes L: u Often easier to maximize lnL. L and lnL are both maximum at the same location. Also, 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 Let’s assume that  is the same for each measurement. u The likelihood function for this problem is: gaussian pdf

R. Kass/W03 P416 Lecture 5 u Find the  that maximizes the log likelihood function: u If  are different for each data point then  is just the weighted average: Average factor out  since it is a constant Weighted Average don’t forget the factor of n

R. Kass/W03 P416 Lecture 5 l Example 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

R. Kass/W03 P416 Lecture 5 Maximum Likelihood Fit of Data to a Function 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, there’s an x which is known exactly. u Suppose we know the functional relationship between the y’s and the x’s: n , ...are parameters. MLM gives us a method to determine , ... from our data. l Example: Fitting data points to a straight line: two linear equations with two unkowns u Find  and  by maximizing the likelihood function L likelihood function:

R. Kass/W03 P416 Lecture 5 For the sake of simplicity assume that all  ’s are the same (  1 =  2 =  ): We now have two equations that are linear in the unknowns ,  : in matrix form These equations correspond to Taylor’s Eq

R. Kass/W03 P416 Lecture 5 l EXAMPLE: u 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

R. Kass/W03 P416 Lecture 5 u The line “best” represents our data. u Not all the data points are "on" the line. uThe line minimizes the sum of squares of the deviations (  ) between the line and our data (d i ):  i =data-prediction= d i -(d 0 +vt i ) The MLM fit to the data for d=d 0 +vt