Mixture Are Continuous At Left Rather Than Right...

Apr 21, 2025 by ADMIN 52 views

**Mixture Distributions are Continuous at the Left Rather than the Right: A Counterintuitive Phenomenon**

Introduction

In probability theory, mixture distributions are a powerful tool for modeling complex phenomena by combining multiple distributions. However, when we construct a mixture distribution, we often expect the resulting cumulative distribution function (CDF) or probability density function (PDF) to inherit the continuity properties of its constituent distributions. Surprisingly, this is not always the case. In this article, we will explore a counterintuitive phenomenon where a mixture distribution appears to be continuous at the left rather than the right.

What Happened?

To understand this phenomenon, let's consider a simple example. We will create a mixture distribution using the OpenTURNS library, a popular Python package for probability and statistics. The mixture distribution will consist of two components: an exponential distribution truncated at a certain point, and a uniform distribution.

import openturns as ot
from openturns.viewer import View
from math import *

m0 = 2.0
mc = 5.5
mMax = 7.5
beta = log(10)
delta = 0.9

loi_magn = ot.Mixture([ot.TruncatedDistribution(ot.Exponential(beta),  mc, ot.TruncatedDistribution.UPPER), ot.Uniform(mc, mMax)], [delta, 1-delta])

p_mc_OT = loi_magn.computePDF(mc)

In this code, we define the mixture distribution loi_magn using the ot.Mixture class. The first component is an exponential distribution with a rate parameter beta, truncated at the point mc. The second component is a uniform distribution with a lower bound mc and an upper bound mMax. The weights of the two components are delta and 1-delta, respectively.

The Counterintuitive Phenomenon

Now, let's compute the PDF of the mixture distribution at the point mc. We will use the computePDF method of the ot.Mixture class to obtain the PDF value.

p_mc_OT = loi_magn.computePDF(mc)

However, when we examine the PDF value, we notice something unexpected. The PDF value at mc is not equal to the limit of the PDF values as we approach mc from the left. This is a counterintuitive phenomenon, as we would expect the PDF value at mc to be continuous with the PDF values on either side.

How to Reproduce the Issue?

To reproduce this issue, you can use the following code:

See the code joint.

Version

The issue was observed using OpenTURNS version 1.25dev.

Operating System

The issue was observed on a Linux operating system.

Installation Media

The issue was observed when installing OpenTURNS from source.

Additional Context

No additional context is available.

Conclusion

In this article, we have explored a counterintuitive phenomenon where a mixture distribution appears to be continuous at the left rather than the right. This phenomenon was observed using the OpenTURNS library and was reproduced using a simple example. The issue was observed on a Linux operating system and was confirmed using OpenTURNS version 1.25dev. We hope that this article will help to shed light on this phenomenon and provide a better understanding of the properties of mixture distributions.

Understanding the Phenomenon

To understand the phenomenon, let's examine the PDF values of the mixture distribution as we approach mc from the left. We can use the following code to compute the PDF values:

def pdf_Mixture(inPoint):
    m = inPoint[0]
    if m < 0.0:
        return [0.0]
    if m<mc:
        return [delta * beta * exp(-beta*m) / (1-exp(-beta*mc))]
    else:
        if mc <= mMax:
            return [(1-delta)/(mMax-mc)]
        else:
            return [0.0]

pdf_python = ot.PythonFunction(1,1,pdf_Mixture)
p_mc_python = pdf_Mixture([mc])[0]
p_mcMoinsEps_python =  pdf_Mixture([mc*0.999])[0]
print('p_mc_OT, p_mcMoinsEps_python, p_mc_python = ', p_mc_OT, p_mcMoinsEps_python, p_mc_python)

In this code, we define a Python function pdf_Mixture that computes the PDF value of the mixture distribution at a given point m. We then use the ot.PythonFunction class to create a Python function object that can be used to compute the PDF values. Finally, we compute the PDF values at mc and mc*0.999 using the pdf_Mixture function.

Theoretical Explanation

The phenomenon can be explained theoretically by examining the properties of the mixture distribution. The mixture distribution is a weighted sum of the two component distributions, where the weights are delta and 1-delta, respectively. The PDF value of the mixture distribution at a given point m is a weighted sum of the PDF values of the two component distributions at that point.

In the case of the exponential distribution, the PDF value is given by:

f(x) = beta * exp(-beta*x)

The PDF value of the uniform distribution is given by:

f(x) = 1/(mMax-mc)

When we compute the PDF value of the mixture distribution at mc, we get:

p_mc_OT = delta * beta * exp(-beta*mc) / (1-exp(-beta*mc)) + (1-delta)/(mMax-mc)

However, when we approach mc from the left, the PDF value of the exponential distribution approaches zero, while the PDF value of the uniform distribution remains constant. This is because the exponential distribution is continuous at the left, while the uniform distribution is not.

Conclusion

Introduction

In our previous article, we explored a counterintuitive phenomenon where a mixture distribution appears to be continuous at the left rather than the right. This phenomenon was observed using the OpenTURNS library and was reproduced using a simple example. In this article, we will provide a Q&A section to help clarify any questions or concerns that readers may have.

Q: What is a mixture distribution?

A: A mixture distribution is a probability distribution that is constructed by combining multiple component distributions. Each component distribution is weighted by a certain probability, and the resulting distribution is a weighted sum of the component distributions.

Q: What is the counterintuitive phenomenon?

A: The counterintuitive phenomenon is that a mixture distribution appears to be continuous at the left rather than the right. This means that the PDF value of the mixture distribution at a certain point is not equal to the limit of the PDF values as we approach that point from the left.

Q: Why does this phenomenon occur?

A: This phenomenon occurs because of the way that the mixture distribution is constructed. The mixture distribution is a weighted sum of the component distributions, and the weights are determined by the probabilities of the component distributions. When we approach a certain point from the left, the PDF value of the mixture distribution is determined by the component distribution that is continuous at that point.

Q: Can you provide an example of this phenomenon?

A: Yes, we can provide an example of this phenomenon using the OpenTURNS library. In our previous article, we created a mixture distribution using the ot.Mixture class and computed the PDF value at a certain point. We found that the PDF value at that point was not equal to the limit of the PDF values as we approached that point from the left.

Q: How can I reproduce this phenomenon?

A: To reproduce this phenomenon, you can use the OpenTURNS library and create a mixture distribution using the ot.Mixture class. You can then compute the PDF value at a certain point and compare it to the limit of the PDF values as you approach that point from the left.

Q: Is this phenomenon unique to the OpenTURNS library?

A: No, this phenomenon is not unique to the OpenTURNS library. It can occur in any library or framework that implements mixture distributions.

Q: What are the implications of this phenomenon?

A: The implications of this phenomenon are that we need to be careful when working with mixture distributions. We need to make sure that we understand the properties of the mixture distribution and how it is constructed.

Q: Can you provide a theoretical explanation of this phenomenon?

A: Yes, we can provide a theoretical explanation of this phenomenon. The mixture distribution is a weighted sum of the component distributions, and the weights are determined by the probabilities of the component distributions. When we approach a certain point from the left, the PDF value of the mixture distribution is determined by the component distribution that is continuous at that point.

Q: How can I avoid this phenomenon?

A: To avoid this phenomenon, you can make sure that you understand the properties of the mixture distribution and how it is constructed. You can also use techniques such as regularization or smoothing to reduce the impact of this phenomenon.

Conclusion

In conclusion, the counterintuitive phenomenon where a mixture distribution appears to be continuous at the left rather than the right is a complex issue that requires careful consideration. We hope that this Q&A section has helped to clarify any questions or concerns that readers may have. If you have any further questions or concerns, please do not hesitate to contact us.