Lecture 2: Parameter Estimation and Evaluation of Support

Parameter Estimation “The problem of estimation is of more central importance, (than hypothesis testing).. for in almost all situations we know that the effect whose significance we are measuring is perfectly real, however small; what is at issue is its magnitude.” (Edwards, 1992, pg. 2) “An insignificant result, far from telling us that the effect is non- existent, merely warns us that the sample was not large enough to reveal it.” (Edwards, 1992, pg. 2)

Parameter Estimation l Finding Maximum Likelihood Estimates (MLEs) - Local optimization ( optim ) »Gradient methods »Simplex (Nelder-Mead) - Global optimization »Simulated Annealing ( annea l) »Genetic Algorithms ( rgenoud ) l Evaluating the strength of evidence (“support”) for different parameter estimates - Support Intervals »Asymptotic Support Intervals »Simultaneous Support Intervals - The shape of likelihood surfaces around MLEs

Parameter estimation: finding peaks on likelihood “surfaces”... The variation in likelihood for any given set of parameter values defines a likelihood “surface”... The goal of parameter estimation is to find the peak of the likelihood surface.... (optimization)

Local vs Global Optimization l “Fast” local optimization methods - Large family of methods, widely used for nonlinear regression in commercial software packages l “Brute force” global optimization methods - Grid search - Genetic algorithms - Simulated annealing local optimum global optimum

Local Optimization – Gradient Methods l Derivative-based (Newton-Raphson) methods: Likelihood surface General approach: Vary parameter estimate systematically and search for zero slope in the first derivative of the likelihood function...(using numerical methods to estimate the derivative, and checking the second derivative to make sure it is a maximum, not a minimum)

Local Optimization – No Gradient l The Simplex (Nelder Mead) method - Much simpler to program - Does not require calculation or estimation of a derivative - No general theoretical proof that it works, (but lots of happy practitioners…) Implemented as method= “Nelder-Mead” in the “optim” function in R

Global Optimization – Grid Searches l Simplest form of optimization (and rarely used in practice) - Systematically search parameter space at a grid of points l Can be useful for visualization of the broad features of a likelihood surface

Global Optimization – Genetic Algorithms l Based on a fairly literal analogy with evolution - Start with a reasonably large “population” of parameter sets - Calculate the “fitness” (likelihood) of each individual set of parameters - Create the next generation of parameter sets based on the fitness of the “parents”, and various rules for recombination of subsets of parameters (genes) - Let the population evolve until fitness reaches a maximum asymptote Implemented in the “genoud” package in R: cool but slow for large datasets with large number of parameters

Global optimization - Simulated Annealing l Analogy with the physical process of annealing: - Start the process at a high “temperature” - Gradually reduce the temperature according to an annealing schedule l Always accept uphill moves (i.e. an increase in likelihood) l Accept downhill moves according to the Metropolis algorithm: p = probability of accepting downhill move  lh = magnitude of change in likelihood t = temperature

Effect of temperature (t)

Simulated Annealing in practice... REFERENCES: Goffe, W. L., G. D. Ferrier, and J. Rogers Global optimization of statistical functions with simulated annealing. Journal of Econometrics 60: Corana et al Minimizing multimodal functions of continuous variables with the simulated annealing algorithm. ACM Transactions on Mathematical Software 13: A version with automatic adjustment of range... Lower boundUpper bound Current value Search range (step size)

Effect of C on Adjusting Range...

Constraints – setting limits for the search... l Biological limits - Values that make no sense biologically (be careful...) l Algebraic limits - Values for which the model is undefined (i.e. dividing by zero...) Bottom line: global optimization methods let you cast your net widely, at the cost of computer time...

Simulated Annealing - Initialization l Set - Annealing schedule »Initial temperature (t) (3.0) »Rate of reduction in temperature (rt) (0.95)N »Interval between drops in temperature (nt) (100) »Interval between changes in range (ns) (20) - Parameter values »Initial values (x) »Upper and lower bounds (lb,ub) »Initial range (vm) Typical values in blue...

How many iterations?... Red maple leaf litterfall (6 parameters) 500,000 is way more than necessary! Logistic regression of windthrow susceptibility (188 parameters) 5 million is not enough! What would constitute convergence?...

Optimization - Summary l No hard and fast rules for any optimization – be willing to explore alternate options. l Be wary of initial values used in local optimization when the model is at all complicated l How about a hybrid approach? Start with simulated annealing, then switch to a local optimization…

Evaluating the strength of evidence for the MLE Now that you have an MLE, how should you evaluate it? (Hint: think about the shape of the likelihood function, not just the MLE)

Strength of evidence for particular parameter estimates – “Support” l Likelihood provides an objective measure of the strength of evidence for different parameter estimates... Log-likelihood = “Support” (Edwards 1992)

Profile Likelihood l Evaluate support (information) for a range of values of a given parameter by treating all other parameters as “nuisance” and holding them at their MLEs… Parameter 1 Parameter 2

Asymptotic vs. Simultaneous M-Unit Support Limits l Asymptotic Support Limits (based on Profile Likelihood): - Hold all other parameters at their MLE values, and systematically vary the remaining parameter until likelihood declines by a chosen amount (m)... What should “m” be? (2 is a good number, and is roughly analogous to a 95% CI)

Asymptotic vs. Simultaneous M-Unit Support Limits l Simultaneous: - Resampling method: draw a very large number of random sets of parameters and calculate log- likelihood. M-unit simultaneous support limits for parameter x i are the upper and lower limits that don’t differ by more than m units of support... In practice, it can require an enormous number of iterations to do this if there are more than a few parameters

Asymptotic vs. Simultaneous Support Limits Parameter 1 Parameter 2 2-unit drop in support A hypothetical likelihood surface for 2 parameters... Asymptotic 2-unit support limits for P1 Simultaneous 2-unit support limits for P1

Other measures of strength of evidence for different parameter estimates l Edwards (1992; Chapter 5) - Various measures of the “shape” of the likelihood surface in the vicinity of the MLE... How pointed is the peak?...

Evaluating Support for Parameter Estimates: A Frequentist Approach l Traditional confidence intervals and standard errors of the parameter estimates can be generated from the Hessian matrix - Hessian = matrix of second partial derivatives of the likelihood function with respect to parameters, evaluated at the maximum likelihood estimates - Also called the “Information Matrix” by Fisher - Provides a measure of the steepness of the likelihood surface in the region of the optimum - Can be generated in R using optim and fdHess

Example from R The Hessian matrix (when maximizing a log likelihood) is a numerical approximation for Fisher's Information Matrix (i.e. the matrix of second partial derivatives of the likelihood function), evaluated at the point of the maximum likelihood estimates. Thus, it's a measure of the steepness of the drop in the likelihood surface as you move away from the MLE. > res$hessian a b sd a b sd (sample output from an analysis that estimates two parameters and a variance term)

More from R now invert (“solve” in R parlance) the negative of the Hessian matrix to get the matrix of parameter variance and covariance > solve(-1*res$hessian) a b sd a e e e-06 b e e e-07 sd e e e-03 the square roots of the diagonals of the inverted negative Hessian are the standard errors* > sqrt(diag(solve(-1*res$hessian))) a b sd (*and 1.96 * S.E. is a 95% C.I….)