Marginal Vs Conditional Distribution: A Comprehensive Comparison

When studying probability theory or statistics, two terms that are commonly brought up are marginal and conditional distribution. Both of these distributions have important roles in analyzing and understanding data, but they approach the distribution in different ways. In this article, we will compare marginal and conditional distributions, explain their differences and similarities, and discuss typical use cases for both.

What is a Marginal Distribution?

A marginal distribution is the probability distribution of one or more variables in a dataset. It is the distribution of a single variable after considering every possible value of all other variables. In other words, given a set of variables X₁, X₂, … Xₙ, their marginal distribution Xᵢ is the probability distribution over the values of Xᵢ, without specifying a value for any of the other variables.

For example, suppose we have the following dataset:

| X | Y |

|—|—|

| 1 | 2 |

| 1 | 5 |

| 2 | 1 |

| 2 | 6 |

The marginal distribution of X in this dataset would be:

| X | Probability |

|—|————|

| 1 | 0.5 |

| 2 | 0.5 |

The probabilities here are calculated by taking the frequency of each X value and dividing by the total number of values, regardless of the value of Y.

What is a Conditional Distribution?

On the other hand, a conditional distribution is the probability distribution of one or more variables in a dataset given the value(s) of other variables. It is a probability distribution of a variable when it is bound by a specific condition(s). It provides the likelihood of the occurrence of an event based on the occurrence of another event. Mathematically, a conditional distribution of variable X given variable Y is written as P(X|Y).

For example, suppose we want to find the conditional distribution of X given Y=5 in the same dataset:

| X | Y |

|—|—|

| 1 | 2 |

| 1 | 5 |

| 2 | 1 |

| 2 | 6 |

The conditional distribution of X given Y=5 can be calculated as follows:

| X | Probability |

|—|————|

| 1 | 0.5 |

| 2 | 0.5 |

Here, the probabilities are calculated by only considering the values of X where Y=5.

Comparison between the Two

Marginal and conditional distributions serve their purposes on their own. However, when compared, they have some significant differences.

Marginal distribution is an unconditional probability distribution, while a conditional distribution is a conditional probability distribution. Marginal distribution is concerned with the probability distribution of one or more variables, irrespective of the other variables in the dataset. On the other hand, conditional distribution is concerned with the probability distribution of one or more variables, given a specific value or set of values of some other variable(s) in the dataset.

Conditional distributions are based on specific criteria, while marginal distributions provide a general idea of the entire dataset. Therefore, conditional distribution is a modification of a marginal distribution, which applies specific criteria to a particular dataset.

Another distinction between the two is how they are calculated. In marginal distribution, we sum or average the values of a dataset using all possible values of each variable. In contrast, conditional distribution requires applying specific logical or arithmetic criteria in calculating the likelihood of a particular event.

Finally, a good way to think about the difference between the two is by considering the target audience. Marginal distribution is ideal for a statistical modeler analyzing the entire dataset. Conditional distribution is ideal for answering a specific question like, what is the probability of X given Y.

Use Cases

Marginal and conditional distributions serve specific purposes, and they are used in different contexts.

Marginal distribution provides a summary of a dataset, which is useful in providing basic statistical information about the dataset. For example, it can be used to determine the average or total of a given variable in a dataset. It is also useful in determining that the probability of an X based on the frequency of its occurrence within the dataset. Marginal distribution is used when dealing with more than one variable because it helps to provide information about each variable separately.

Conditional distribution is used when you know something specific about a data set. It is used in making inferences that depend on restricted criteria of a dataset. For example, if one wants to know the probability of rain given that the temperature is above 70 degrees, a conditional distribution is useful. This tells us the likelihood of the occurrence of one event given that the other event has occurred.

Conditional distribution is also useful in machine learning applications, where algorithms learn decision rules based on the relationship between input variables and output variables. A great example of this is understanding why an individual might buy a particular product online. Here, we use demographic data to predict user behavior.

Frequently Asked Questions

1. Why are marginal and conditional distributions important?

Marginal and conditional distributions provide success criteria in describing probability distributions. Marginal distribution is used when dealing with more than one variable because it helps to provide information about each variable separately. conditional distribution is used when you know something specific about a data set. It is used in making inferences that depend on restricted criteria of a dataset.

2. What is the difference between marginal distribution and conditional distribution?

On one hand, who think of marginal distribution thinks of an unconditional probability distribution. On the other hand, a conditional distribution is a conditional probability distribution. Marginal distribution provides information regarding one or more variables irrespective of the other variables in the dataset. In contrast, conditional distribution is concerned with the probability distribution of one or more variables, given a specific value or set of values of some other variable(s) in the dataset.

3. Can you derive one from the other?

Conditional distribution is a modification of a marginal distribution, which applies specific criteria to a particular dataset. Therefore, you cannot derive one from the other. However, given a marginal distribution and a criterion, we can derive a conditional distribution.

4. What are some use cases of marginal distribution and conditional distribution?

Marginal distribution is useful in providing basic statistical information about the dataset, like determining the average, total, or probability of a variable within a dataset. Conditional distribution is used when you want to know something specific about a data set, like the probability of an event occurring given that another event has happened.

5. What is a good analogy for explaining the differences between marginal and conditional distribution?

A good way to think about the difference between the two is by considering the target audience. Marginal distribution is ideal for a statistical modeler analyzing the entire dataset. Conditional distribution is ideal for answering a specific question like, what is the probability of X given Y.

Conclusion

Marginal and conditional distributions both have critical roles in understanding and analyzing data in statistics and probability theory. Marginal distribution provides basic statistical information about the dataset, while conditional distribution gives the likelihood of an event given that another event has occurred. In terms of calculation, marginal distribution involves summing or averaging values of a dataset, while conditional distribution requires applying specific logical or arithmetic criteria in computing the probability of an event. These two distributions serve different purposes, depending on what dataset one is working on and what questions they are trying to answer.