Maximum likelihood (ML) and likelihood ratio (LR) test • Conditional distribution and likelihood • Maximum likelihood estimator • Information in the data and likelihood • Observed and Fisher’s information • Likelihood ratio test • Exercise
Introduction It is often the case that we are interested in finding values of some parameters of the system. Then we design an experiment and get some observations (x1,,,xn). We want to use these observation and estimate parameters of the system. Once we know (it may be a challenging mathematical problem) how parameters and observations are related then we can use this fact to estimate parameters. Maximum likelihood is one the techniques to estimate parameters using observations or experiments. There are other estimation techniques also. These include Bayesian, least-squares, method of moments, minimum chi-squared. The result of the estimation is a function of observation – t(x1,,,xn). It is a random variable and in many cases we want to find its distribution also. In general, finding the distribution of the statistic is a challenging problem. But there are numerical technique to deal with this.
Desirable properties of estimation • Unbiasedness. Bias is defined as difference between estimator (t) and true parameter (). Expectation taken using probability distribution of observations • Efficiency. Efficient estimation is that with minimum variance (var(t)). • Consistency. As the number of observation goes to infinity an estimator converges to true value • Minimum mean square error (m.s.e). M.s.e. is defined as the expectation value of the square of the difference between estimator and the true value It means that this estimator must be efficient and unbiased. It is very difficult to achieve all these properties. Under some conditions ML estimator obeys them asymptotically. Moreover ML is asymptotically normal that simplifies the interpretation of results.
Conditional probability distribution and likelihood Let us assume that we know that our random sample points came from the the population with the distribution with parameter(s) . We do not know . If we would know it then we could write the probability distribution of a single observation f(x|). Here f(x|) is the conditional distribution of the observed random variable if the parameter would be known. If we observe n independent sample points from the same population then the joint conditional probability distribution of all observations can be written: We could write the product of the individual probability distribution because the observations are independent (independent conditionally when parameters are known). f(x|) is the probability of an observation for discrete and density of the distribution for continuous cases. We could interpret f(x1,x2,,,xn|) as the probability of observing given sample points if we would know parameter . If we would vary the parameter(s) we would get different values for the probability f. Since f is the probability distribution, parameters are fixed and observation varies. For a given observation we define likelihood proportional to the conditional probability distribution.
Conditional probability distribution and likelihood: Cont. When we talk about conditional probability distribution of the observations given parameter(s) then we assume that parameters are fixed and observations vary. When we talk about likelihood then observations are fixed parameters vary. That is the major difference between likelihood and conditional probability distribution. Sometimes to emphasize that parameters vary and observations are fixed, likelihood is written as: In this and following lectures we will use one notation for probability and likelihood. When we will talk about probability then we will assume that observations vary and when we will talk about likelihood we will assume that parameters vary. Principle of maximum likelihood states that best parameters are those that maximise probability of observing current values of observations. Maximum likelihood chooses parameters that satisfy:
Maximum likelihood Purpose of maximum likelihood is to maximize the likelihood function and estimate parameters. If derivatives of the likelihood function exist then it can be done using: Solution of this equation will give possible values for maximum likelihood estimator. If the solution is unique then it will be the only estimator. In real application there might be many solutions. Usually instead of likelihood its logarithm is maximized. Since log is strictly monotonically increasing function, derivative of the likelihood and derivative of the log of likelihood will have exactly same roots. If we use the fact that observations are independent then joint probability distributions of all observations is equal to product of individual probabilities. We can write log of the likelihood (denoted as l): Usually working with sums is easier than working with products
Maximum likelihood: Example – success and failure Let us consider two examples. First example corresponds to discrete probability distribution. Let us assume that we carry out trials. Possible outcomes of the trials are success or failure. Probability of success is and probability of failure is 1- . We do not know value of . Let us assume we have n trials and k of them are successes and n-k of them are failures. Value of random variable describing our trials are either 0 (failure) or 1 (success). Let us denote observations as y=(y1,y2,,,,yn). Probability of the observation yi at the ith trial is: Since individual trials are independent we can write for n trials: For log of this function we can write: Derivative of the likelihood w.r.t unknown parameter is: The ML estimator for the parameter is equal to the fraction of successes.
Maximum likelihood: Example – success and failure In the example of successes and failures the result was not unexpected and we could have guessed it intuitively. More interesting problems arise when parameter itself becomes function of some other parameters and possible observations also. Let us say: It may happen that xi themselves are random variables also. If it is the case and the function corresponds to normal distribution then analysis is called Probit analysis. Then log likelihood function would look like: Finding maximum of this function is more complicated. This problem can be considered as a non-linear optimization problem. This kind of problems are usually solved iteratively. I.e. a solution to the problem is guessed and then it is improved iteratively.
Maximum likelihood: Example – normal distribution Now let us assume that the sample points came from the population with normal distribution with unknown mean and variance. Let us assume that we have n observations, y=(y1,y2,,,yn). We want to estimate the population mean and variance. Then log likelihood function will have the form: If we get derivative of this function w.r.t mean value and variance then we can write: Fortunately first of these equations can be solved without knowledge about the second one. Then if we use result from the first solution in the second solution (substitute by its estimate) then we can solve second equation also. Result of this will be sample variance:
Maximum likelihood: Example – normal distribution Maximum likelihood estimator in this case gave sample mean and biased sample variance. Many statistical techniques are based on maximum likelihood estimation of the parameters when observations are distributed normally. All parameters of interest are usually inside mean value. In other words is a function of several parameters. Then problem is to estimate parameters using maximum likelihood estimator. Usually either x-s are fixed values (fixed effects model) or random variables (random effects model). Parameters are -s. If this function is linear on parameters then we have linear regression. If variances are known then the Maximum likelihood estimator using observations with normal distribution becomes least-squares estimator.
Information matrix: Observed and Fisher’s One of the important aspects of the likelihood function is its behavior near to the maximum. If the likelihood function is flat then observations have little to say about the parameters. It is because changes of the parameters will not cause large changes in the probability. That is to say same observation can be observed with similar probabilities for various values of the parameters. On the other hand if likelihood has a pronounced peak near to the maximum then small changes of the parameters would cause large changes in the probability. In this cases we say that observation has more information about parameters. It is usually expressed as the second derivative (or curvature) of the minus log-likelihood function. Observed information is equal to the second derivative of the minus log-likelihood function: When there are more than one parameter it is called information matrix. Usually it is calculated at the maximum of the likelihood. There are other definitions of information also. Example: In case of successes and failures we can write:
Information matrix: Observed and Fisher’s Expected value of the observed information matrix is called expected information matrix or Fisher’s information. Expectation is taken over observations: It is calculated at any value of the parameter. Remarkable fact about Fisher’s information matrix is that it is also equal to the expected value of the product of the gradients (first derivatives): Note that observed information matrix depends on particular observation whereas expected information matrix depends only on the probability distribution of the observations (It is a result of integration. When we integrate over some variables we loose dependence on particular values): When sample size becomes large then maximum likelihood estimator becomes approximately normally distributed with variance close to : Fisher points out that inversion of observed information matrix gives slightly better estimate to variance than that of the expected information matrix.
Information matrix: Observed and Fisher’s More precise relation between expected information and variance is given by Cramer and Rao inequality. According to this inequality variance of the maximum likelihood estimator never can be less than inversion of information: Now let us consider an example of successes and failures. If we get expectation value for the second derivative of minus log likelihood function we can get: If we take this at the point of maximum likelihood then we can say that variance of the maximum likelihood estimator can be approximated by: This statement is true for large sample sizes.
Likelihood ratio test Let us assume that we have a sample of size n (x=(x1,,,,xn)) and we want to estimate a parameter vector =(1,2). Both 1 and 2can also be vectors. We want to test null-hypothesis against alternative one: Let us assume that likelihood function is L(x| ). Then likelihood ratio test works as follows: 1) Maximise the likelihood function under null-hypothesis (I.e. fix parameter(s) 1 equal to 10 , find the value of likelihood at the maximum, 2)maximise the likelihood under alternative hypothesis (I.e. unconditional maximisation), find the value of the likelihood at the maximum, then find the ratio: w is the likelihood ratio statistic. Tests carried out using this statistic are called likelihood ratio tests. In this case it is clear that: If the value of w is small then null-hypothesis is rejected. If g(w) is the the density of the distribution for w then critical region can be calculated using:
References • Berthold, M. and Hand, DJ (2003) “Intelligent data analysis” • Stuart, A., Ord, JK, and Arnold, S. (1991) Kendall’s advanced Theory of statistics. Volume 2A. Classical Inference and the Linear models. Arnold publisher, London, Sydney, Auckland
Exercise 1 a) Assume that we have a sample of size n independently drawn from the population with the density of probability (exponential distribution) What is the maximum likelihood estimator for . What is the observed and expected information. b) Let us assume that we have a sample of size n of two-dimensional vectors ((x1,x2)=((x11,x21),(x12,x22),,,,(x1n,x2n) from the normal distribution: Find the maximum of the likelihood under the following hypotheses: Try to find the likelihood ratio statistic. Note that variance is also unknown.