Glossary

5.3 Marginal distributions

4 min read•august 20, 2024

Marginal distributions simplify complex probability scenarios by focusing on a single variable within a multivariate distribution. They're obtained by summing or integrating joint distributions, providing a streamlined view of individual variable probabilities.

Understanding marginal distributions is crucial for analyzing multivariate data and making informed decisions. They're widely used in , machine learning, and statistical modeling, helping to extract key insights from complex probability distributions.

Definition of marginal distributions

  • Marginal distributions focus on the probability distribution of a single random variable within a multivariate probability distribution
  • Obtained by summing or integrating the joint probability distribution over the values of the other random variables
  • Provides a simplified view of the probability distribution of a specific variable, disregarding the relationships with other variables

Marginal probability mass functions

Discrete random variables

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  • For discrete random variables, the (PMF) is calculated by summing the joint PMF over the values of the other random variables
  • Given a joint PMF P(X=x,Y=y)P(X=x, Y=y), the marginal PMF of XX is P(X=x)=∑yP(X=x,Y=y)P(X=x) = \sum_{y} P(X=x, Y=y)
  • Similarly, the marginal PMF of YY is P(Y=y)=∑xP(X=x,Y=y)P(Y=y) = \sum_{x} P(X=x, Y=y)
  • Marginal PMFs represent the probability distribution of a single discrete random variable, ignoring the values of other variables

Marginal probability density functions

Continuous random variables

  • For continuous random variables, the (PDF) is obtained by integrating the joint PDF over the values of the other random variables
  • Given a joint PDF f(x,y)f(x, y), the marginal PDF of XX is fX(x)=∫−∞∞f(x,y)dyf_X(x) = \int_{-\infty}^{\infty} f(x, y) dy
  • Similarly, the marginal PDF of YY is fY(y)=∫−∞∞f(x,y)dxf_Y(y) = \int_{-\infty}^{\infty} f(x, y) dx
  • Marginal PDFs describe the probability density of a single continuous random variable, disregarding the values of other variables

Marginal distributions vs joint distributions

  • Joint distributions capture the complete probability distribution of multiple random variables, considering their relationships and dependencies
  • Marginal distributions focus on the probability distribution of a single random variable, ignoring the relationships with other variables
  • Marginal distributions can be derived from joint distributions, but joint distributions cannot always be reconstructed from marginal distributions alone
  • Understanding both joint and marginal distributions is crucial for analyzing multivariate probability distributions and their properties

Computing marginal distributions

From joint probability mass functions

  • To compute the marginal PMF of a discrete random variable, sum the joint PMF over the values of the other random variables
  • Example: Given a joint PMF P(X=x,Y=y)P(X=x, Y=y), calculate P(X=x)P(X=x) by summing P(X=x,Y=y)P(X=x, Y=y) over all possible values of YY
  • The resulting marginal PMF represents the probability distribution of the selected random variable

From joint probability density functions

  • To compute the marginal PDF of a continuous random variable, integrate the joint PDF over the values of the other random variables
  • Example: Given a joint PDF f(x,y)f(x, y), calculate fX(x)f_X(x) by integrating f(x,y)f(x, y) with respect to yy over its entire range
  • The resulting marginal PDF describes the probability density of the selected random variable

Properties of marginal distributions

Expectation and variance

  • The (mean) of a can be calculated using the marginal PMF or PDF
  • For discrete random variables, E[X]=∑xx⋅P(X=x)E[X] = \sum_{x} x \cdot P(X=x)
  • For continuous random variables, E[X]=∫−∞∞x⋅fX(x)dxE[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) dx
  • The of a marginal distribution can also be computed using the marginal PMF or PDF
  • Marginal expectations and variances provide insights into the central tendency and dispersion of individual random variables

Relationship to conditional distributions

  • Marginal distributions and conditional distributions are related through the concept of conditional probability
  • The of a random variable given the value of another variable can be obtained by dividing the by the marginal distribution of the conditioning variable
  • Marginal distributions can be derived from conditional distributions using the law of total probability
  • Understanding the relationship between marginal and conditional distributions is essential for probabilistic reasoning and inference

Applications of marginal distributions

Bayesian inference

  • Marginal distributions play a crucial role in Bayesian inference, where they are used to compute posterior probabilities
  • In Bayesian inference, the (evidence) is calculated by integrating the joint probability distribution over the parameter space
  • Marginal distributions help in model comparison and selection by providing a way to evaluate the evidence for different models
  • Bayesian inference relies on marginal distributions to update beliefs and make probabilistic predictions

Machine learning algorithms

  • Marginal distributions are utilized in various machine learning algorithms, particularly in
  • In , marginal distributions of features are used to compute class-conditional probabilities and make predictions
  • Marginal distributions are employed in expectation-maximization (EM) algorithms for parameter estimation in latent variable models
  • Understanding marginal distributions is essential for developing and implementing probabilistic machine learning techniques

Key Terms to Review (24)

∫: The symbol ∫ represents the integral in mathematics, which is a fundamental concept in calculus and is used to find areas under curves, among other applications. In the context of probability and statistics, integrals are crucial for calculating probabilities, expectations, and marginal distributions from joint distributions, thereby connecting the continuous nature of random variables to their discrete properties.
Bayesian inference: Bayesian inference is a statistical method that uses Bayes' theorem to update the probability of a hypothesis as more evidence or information becomes available. This approach combines prior beliefs with new data to calculate a posterior probability, allowing for more dynamic and flexible statistical modeling. It emphasizes the importance of prior distributions and how they can influence the results of statistical analyses.
Conditional distribution: Conditional distribution refers to the probability distribution of a random variable given that another variable is known to take on a specific value. This concept helps in understanding the relationship between variables and plays a crucial role in analyzing situations where certain conditions or events affect the probabilities of outcomes, making it a vital tool in probability theory and statistics.
Data analysis: Data analysis is the process of inspecting, cleaning, transforming, and modeling data to discover useful information, inform conclusions, and support decision-making. It involves various techniques and methods to summarize the main characteristics of data, uncover patterns, and identify relationships between variables, which are critical in understanding variance, standard deviation, and marginal distributions.
Expectation: Expectation is a fundamental concept in probability and statistics that represents the average value or mean of a random variable, calculated by weighing each possible outcome by its probability. It provides insight into the center of a probability distribution, indicating where the values tend to cluster. Understanding expectation is crucial for making predictions and informed decisions based on random processes.
Expectation-Maximization Algorithm: The expectation-maximization (EM) algorithm is a statistical technique used for finding maximum likelihood estimates of parameters in models with latent variables. This algorithm alternates between estimating the expected value of the latent variables (the expectation step) and optimizing the parameters to maximize the likelihood (the maximization step). This process allows for efficient handling of incomplete data and is particularly useful when dealing with marginal distributions, as it helps to uncover hidden structures in data.
Independence: Independence refers to the concept where the occurrence of one event does not influence the probability of another event occurring. In probability and statistics, understanding independence is crucial because it allows for the simplification of complex problems, especially when working with multiple variables and their relationships, such as marginal and conditional distributions, joint probability density functions, and random variables.
Integrating Probabilities: Integrating probabilities involves the process of calculating the total probability from a joint probability density function across specific intervals or regions. This concept is crucial for understanding how probabilities are distributed over continuous random variables and helps in determining marginal distributions, which isolate the probability of a subset of variables while accounting for others.
Joint Distribution: Joint distribution refers to the probability distribution that captures the likelihood of two or more random variables occurring together. This concept is crucial for understanding how different random variables interact with each other, providing insights into their independence, as well as enabling the computation of marginal and conditional distributions.
Marginal Distribution: Marginal distribution refers to the probability distribution of a subset of variables in a multivariate dataset, obtained by summing or integrating over the other variables. This concept is essential for understanding how individual variables behave within the context of joint distributions, which consider multiple variables simultaneously. It connects to independence, as knowing the marginal distributions can help determine if two variables are independent by checking if their joint distribution equals the product of their marginals.
Marginal Histograms: Marginal histograms are graphical representations that show the distribution of individual variables in a joint distribution, displayed along the axes of a two-dimensional histogram. They help in understanding the behavior of each variable separately, while still being part of the larger context of the joint distribution. These histograms can reveal important characteristics such as skewness, modality, and outliers in the data, providing insights into the relationship between the variables.
Marginal Likelihood: Marginal likelihood refers to the probability of observing the data under a specific model, integrating over all possible values of the model parameters. It plays a critical role in Bayesian statistics, allowing for model comparison by calculating how well different models explain the observed data, regardless of the complexity of the models themselves.
Marginal Probability: Marginal probability refers to the probability of an event occurring without consideration of any other events. It’s derived from the joint probabilities of multiple events, essentially summarizing the likelihood of a single event across all possible scenarios. This concept is crucial in understanding how probabilities can change when we consider additional information, particularly in the context of multiple variables and their interactions.
Marginal Probability Density Function: A marginal probability density function describes the probability distribution of a single random variable from a joint distribution of multiple random variables. It is derived by integrating the joint probability density function over the other variables, thus providing insight into the behavior of one variable independently of the others.
Marginal probability mass function: A marginal probability mass function (PMF) describes the probability distribution of a subset of random variables by summing over the probabilities of all other variables in a joint distribution. This function helps in understanding the behavior of one variable without considering the influence of others, allowing for simpler analysis and interpretation of discrete random variables in multivariate contexts.
Naive bayes classifiers: Naive Bayes classifiers are a family of probabilistic algorithms based on Bayes' theorem, used for classification tasks. They assume that the features are independent given the class label, simplifying the computation of conditional probabilities. This independence assumption makes them 'naive,' but despite this simplicity, they often perform surprisingly well in practice, particularly with large datasets and text classification problems.
P(a): p(a) represents the probability of event A occurring within a given sample space. This notation helps to quantify uncertainty and is foundational to understanding how likely events are to happen. Knowing p(a) is crucial for making informed predictions and decisions based on statistical data.
Probabilistic Graphical Models: Probabilistic graphical models are a powerful framework that represents complex distributions through graphs, combining probability theory and graph theory. They consist of nodes that represent random variables and edges that capture the dependencies between them, allowing for efficient computation of marginal distributions and other probabilistic queries. By visualizing these relationships, they help in understanding the structure of uncertainty in data.
Relationship between marginal and joint probabilities: The relationship between marginal and joint probabilities highlights how individual probabilities of events connect to the probability of their combined occurrence. Marginal probabilities refer to the likelihood of a single event happening, irrespective of other events, while joint probabilities quantify the likelihood of two or more events happening simultaneously. Understanding this relationship is crucial for grasping concepts like independence and conditional probabilities.
Risk Assessment: Risk assessment is the systematic process of evaluating potential risks that may be involved in a projected activity or undertaking. It helps in quantifying the likelihood of adverse events and their potential impact, making it crucial for informed decision-making in uncertain environments.
Scatterplots: A scatterplot is a graphical representation that uses dots to display the values of two different variables, showing how they may relate to each other. Each dot represents an observation in a dataset, with one variable plotted along the x-axis and the other along the y-axis. Scatterplots help visualize relationships, trends, and potential correlations between the two variables, making them essential for analyzing marginal distributions.
Summing probabilities: Summing probabilities refers to the process of adding together the probabilities of individual outcomes within a given sample space to determine the overall probability of certain events. This concept is crucial when dealing with marginal distributions, as it helps in extracting the probabilities associated with specific variables by combining relevant probabilities from joint distributions. By summing probabilities, one can analyze data more effectively and understand how different events relate to each other.
Variance: Variance is a statistical measure that represents the degree of spread or dispersion of a set of values around their mean. It provides insight into how much individual data points differ from the average, helping to understand the distribution of values in both discrete and continuous random variables.
σ (sigma): In statistics, σ represents the standard deviation of a population, which measures the amount of variation or dispersion in a set of values. A smaller value of σ indicates that the values tend to be closer to the mean, while a larger σ indicates that the values are spread out over a wider range. Understanding σ is crucial in interpreting variance and helps in assessing how individual data points relate to the overall distribution.
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