MULTIDIMENSIONAL RARE EVENT PROBABILITY

ESTIMATION ALGORITHM

Ingrida Vaičiulytė

Vilnius UniversityMathematics and Informatics Institute

COMPUTER DAYS – 2013Šiauliai

Introduction

This work describes the empirical Bayesian approach applied in the estimation of multi – dimensional frequency. It also introduces the Monte-Carlo Markov Chain (MCMC) procedure, which is designed for Bayesian computation. Modeling of the discrete variable - the number of occurrences of rare, used statistical models: a normal distribution with unknown parameters - mean and variance, and Poisson distribution.

COMPUTER DAYS – 2013Šiauliai

Introduction

Let us consider a set of K populations, where each populationconsists of individuals Assume that some event (e.g., death due to some disease, insured event) can occur in the populations under observation.

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COMPUTER DAYS – 2013Šiauliai

The aim

Our aim is to estimate unknown probabilities of events when the numbers of events in populations are observedSince a simple estimate of relative risk cannot be used in many cases due to great differences in the population sizethe empirical Bayesian approach is applied.

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COMPUTER DAYS – 2013Šiauliai

Poisson-Gaussian model

An assumption is often justified that the numbers of cases follow to the Poisson distribution with the parametersand its density is as follows:

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COMPUTER DAYS – 2013Šiauliai

Poisson-Gaussian model

The empirical Bayesian method is a two stage procedure, depending on the prior distribution introduced in the second stage. It is of interest to consider a model in which the logits

are normally distributed with the parameters

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COMPUTER DAYS – 2013Šiauliai

Poisson-Gaussian model

Thus the density of logit is

Then the rates are evaluated as a posteriori means for given

where

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COMPUTER DAYS – 2013Šiauliai

Maximum likelihood method

The Bayesian analysis is often related in statistics to the minimization of a certain function, expressed as the integral of a posteriori density. Thus, in the empirical Bayesian approach, the unknown parameters are estimated by the maximum likelihood method.We get the logarithmic likelihood function after some manipulation such as

which have to be minimized to get estimates for the parameters.

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COMPUTER DAYS – 2013Šiauliai

Derivatives of the maximum likelihood function

Likelihood function is differentiable many times with respect to the parameters and the respective first derivatives of this function are as follows:

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COMPUTER DAYS – 2013Šiauliai

Poisson-Gaussian model estimates

The maximum likelihood estimates of parameters of Poisson-Gaussian model are found by solving equations, where the first derivatives must be equal to zero:

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COMPUTER DAYS – 2013Šiauliai

Poisson-Gaussian model estimates

For instance, the “fixed point iteration” method is useful to solve these equations in order to get the maximum likelihood estimates of :,

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

The “fixed point iteration” method we can to realize by Monte-Carlo Markov chain approach. Let be generated t chains and in each chain we generate a multivariate Gaussian vector

is the Monte – Carlo sample size at the step.

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

In order to avoid computational problems, when the intermediate results are very small, we have introduced the auxiliary function

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

And then we get estimates of parameters

where the Monte-Carlo estimators are as follows

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

Next, the estimate of the log-likelihood function is obtained using the Monte-Carlo estimate:

its sample variance estimate:

population of events probabilities estimate:

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

The Monte-Carlo chain can be terminated at the step, if difference between estimates of

two current steps differs insignificantly. Thus, the hypothesis on the termination condition is rejected, if

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

The next rule of sample size regulation is implemented; in order large samples would be taken only at the moment of making the decision on termination of the Monte-Carlo Markov chain:

- Fisher’s quantile, - is the significance level.

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COMPUTER DAYS – 2013Šiauliai

MCMC algorithm

Application of this rule allows to rational select of samples size in Monte-Carlo Markov chain to ensure the convergence of the maximum likelihood function.

COMPUTER DAYS – 2013Šiauliai

Computer simulation

Next, we used familiar data to construct and estimate this statistical model.The random sample of populations has been simulated to explore the approach developed, in which can occur events. The logits of probabilities are normally distributed with these parameters

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COMPUTER DAYS – 2013Šiauliai

Computer simulation

Next, we have computed the Monte-Carlo Markov chain of estimators. To avoid very small or very large sample sizes, the following limits were applied

The termination conditions started to be valid after iterations.And we have got these means of parameters:

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COMPUTER DAYS – 2013Šiauliai

Estimates of parametersIteration µ1 µ2 µ3

Log-likelihood function

Confidence interval

Sample size

Statistical hypothesis

1 -2,96 -4,29 -5,52 -62,90 5,57 500 9,55

2 -2,89 -4,04 -5,27 -396,58 4,81 500 6,18

3 -2,91 -4,03 -5,19 -420,42 2,97 500 3,86

4 -2,90 -4,04 -5,16 -424,87 3,2 500 0,35

5 -2,91 -4,04 -5,13 -428,05 1,57 2 963 1,41

6 -2,90 -4,04 -5,14 -427,57 1,32 4 383 0,32

7 -2,91 -4,04 -5,13 -425,54 0,75 13 986 0,40

8 -2,91 -4,04 -5,14 -425,33 0,75 14 345 0,40

9 -2,91 -4,04 -5,13 -425,71 0,75 13 525 0,84

10 -2,91 -4,04 -5,13 -426,47 0,75 15 135 0,22

COMPUTER DAYS – 2013Šiauliai

Conclusions

The empirical Bayesian approach applied in the estimation of multi-dimensional frequency has been described in this work. In this paper we:• presented an iterative method of “fixed point iteration” to

compute the estimates;• introduced the Monte-Carlo Markov Chain procedure with

adaptive regulation sample size and treatment of the simulation error in the statistical manner;

• computed the empirical Bayesian estimation of unknown parameters and probabilities of the events.

The approach developed can be applied in the analysis of social and medical data.

COMPUTER DAYS – 2013Šiauliai

Thank you!

COMPUTER DAYS – 2013Šiauliai