Understanding Fisher Information for Poisson Distributed Data

Understanding Fisher Information for Poisson Distributed Data

Poisson distribution is widely used in many fields, including biology, physics, economics, and telecom. In this article, we will delve deep into the concept of Fisher information and explore how it can be applied to Poisson distributed data.

What is Fisher Information?

Fisher information is a key concept in the field of statistical inference. It measures how much information a set of data carries about a particular parameter that we are interested in. In other words, Fisher information gives us an idea of how precisely we can estimate the parameter based on the available data.

How is Fisher Information Calculated?

The Fisher information of a statistical model can be calculated by taking the derivative of the log-likelihood function with respect to the parameter of interest. For Poisson distributed data, the log-likelihood function is given by:

where x_i is the i-th observation, θ is the unknown parameter (in this case, the mean of the Poisson distribution), and n is the sample size.

Taking the derivative of the log-likelihood function with respect to θ, we get:

This gives us the Fisher information for Poisson distributed data, which is equal to 1/θ.

Interpreting Fisher Information

Fisher information has several interesting properties that make it a useful tool in statistical inference. One of the most important properties of Fisher information is that it is a lower bound on the variance of any unbiased estimator of the parameter of interest. In other words, Fisher information tells us how well we can estimate the parameter even in the best-case scenario.

Another important property of Fisher information is that it is additive. This means that if we have a sample of size n that is divided into k sub-samples of sizes n₁, n₂, …, n_k, then the Fisher information for the overall sample is equal to the sum of the Fisher information for each sub-sample.

Example

Suppose we want to estimate the mean number of defects in a manufacturing process. We take a sample of size n=100 and find that the number of defects follows a Poisson distribution with a mean of θ=5.

The Fisher information for the sample is given by:

This means that even in the best-case scenario, the variance of our estimator would be at least 1/0.2 = 5. This also means that if we were to divide our sample into two sub-samples of sizes n₁=50 and n₂=50, then the Fisher information for the overall sample would be equal to:

This shows that the more data we have, the more precise our estimate of the parameter will be, and the higher the Fisher information.

Conclusion

Fisher information is a powerful tool in statistical inference that allows us to estimate the precision of our estimates of the parameter of interest. For Poisson distributed data, the Fisher information is equal to 1/θ, and it tells us how well we can estimate the mean of the distribution given the available data.

The properties of Fisher information, including its additivity and lower bound on the variance of any unbiased estimator, make it an essential concept for anyone working with statistical data. By understanding Fisher information, you can make more precise and accurate estimates of the parameters that you are interested in.