Writing a likelihood function

Multiplying thousands of probabilities together is simply not a viable approach without infinite precision. Model comparison tables based on ICs are now common in the literature, so it is worthwhile to be able to produce your own using the basic likelihood approaches above.

You should also explore nlminb. One issue is that of restrictions upon parameters. All through this, we will use the "ordinary least squares" OLS model a. In addition, several common distributions have likelihood functions that contain products of factors involving exponentiation.

Log-likelihood[ edit ] For many applications, the natural logarithm of the likelihood function, called the log-likelihood, is more convenient to work with. That illustrates an important aspect of likelihoods: These are called Akaike weights wiand are unique to a particular set of models that is, if a model is added or removed, wi must be recalculated.

You might find it convenient to snarf a tarfile of all the. It does not need you to write the gradient. Writing the likelihood function You have to write an R function which computes out the likelihood function.

It is the fastest There are two powerful optimisers in R: There could not be a simpler task for a maximisation routine. Roll your own likelihood function with R This document assumes you know something about maximum likelihood estimation.

It helps you get going in terms of doing MLE in R. In my toy experiment, this seems to be merely a question of speed - using the analytical gradient makes the MLE go faster. This variant uses the log transformation in order to ensure that sigma is positive.

Here we can write a simple function that, given a set of candidate models, creates the standard information-theoretic output: The logarithm of this product is a sum of individual logarithms, and the derivative of a sum of terms is often easier to compute than the derivative of a product.

Analytical derivatives are used. As always in R, this can be done in several different ways. And, which is the fastest? For a Bernoulli variable, this is simply a search through the space of values for p i. One traditional way to deal with this is to "transform the parameter space".

First, we want to define a function that specifies the probability of our entire data set. In such a situation, the likelihood function factors into a product of individual likelihood functions. The smallness of the objective for large problems can become a major problem. Examining the output of optimize, we can see that the likelihood of the data set was maximized very near 0.

Do they all give the same answer? This gives the result: The general algorithm requires that you specify a more general log likelihood function analogous to the R-like pseudocode below: When the search algorithm is running, it may stumble upon nonsensical values - such as a sigma below 0 - and you do need to think about this.

The maximum likelihood approach says that we should select the parameter that makes the data most probable. But the OLS likelihood is unique and simple; it is globally quasiconcave and has a clear top.

SANN This is a stochastic search algorithm based on simulated annealing. Here are the formulae for the OLS likelihood, and the notation that I use. A quick examination of the likelihood function as a function of p makes it clear that any decent optimization algorithm should be able to find the maximum: It is very costly.

This suggests that the optimization approximation can work. This is because we are generally interested in where the likelihood reaches its maximum value: We assume that each observation in the data is independently and identically distributed, so that the probability of the sequence is the product of the probabilities of each value.Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site.

How to calculate the likelihood function. up vote 9 down vote favorite. 7. The likelihood function of a sample, is the joint density of the random variables involved but viewed as a function of the unknown parameters given a specific sample of realizations from these random variables. Algebraically, the likelihood L(θ ; x) is just the same as the distribution f(x; θ), but its meaning is quite different because it is regarded as a function of θ rather than a function of x.

Consequently, a graph of the likelihood usually looks very different from a graph of the probability distribution. I'm trying to estimate the parameters of my Log-Likelihood function given a set of constraints and using the Newton-Raphson method.

My actual target function is more complex than the one that I w. I need to write a likelihood function to be optimized via fmincon. The problem here is that: 1) To simplify things, my likelihood function is dependent on \alpha, \beta where \beta is specified somewhere in the code before the fmincon part.

\alpha is the vector to be estimated via fmincon. 2) I plan. Writing the likelihood function You have to write an R function which computes out the likelihood function.

As always in R, this can be done in several different ways.

Writing a likelihood function
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