1.4 Expectation and variance

Expectation and variance are fundamental concepts in Bayesian statistics, providing tools to analyze random variables and quantify uncertainty. These measures help us understand the average behavior and spread of probability distributions, forming the basis for parameter estimation and prediction in Bayesian inference.

Expectation calculates the average outcome, while variance measures the spread around that average. Together, they enable us to characterize distributions, make informed decisions, and update our beliefs as we gather new data. These concepts are essential for understanding the core principles of Bayesian analysis and their practical applications.

Definition of expectation

• Expectation quantifies the average outcome of a random variable in probability theory and statistics
• Plays a crucial role in Bayesian statistics for estimating parameters and making predictions based on prior knowledge and observed data

Probability-weighted average

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• Calculates the sum of all possible values multiplied by their respective probabilities
• Expressed mathematically as $E[X] = \sum_{i=1}^{n} x_i \cdot p(x_i)$ for discrete random variables
• Represents the center of mass of a probability distribution
• Used to determine the long-run average outcome of repeated experiments

Discrete vs continuous cases

• Discrete case involves summing over finite or countably infinite possible values
• Continuous case requires integration over the entire range of the random variable
• Continuous expectation formula: $E[X] = \int_{-\infty}^{\infty} x \cdot f(x) dx$, where f(x) is the probability density function
• Both cases yield a single value representing the average outcome
• Provides a foundation for comparing different probability distributions

Properties of expectation

• Expectation serves as a fundamental tool in probability theory and Bayesian statistics
• Enables the analysis of random variables' behavior and relationships between multiple variables

Linearity of expectation

• States that the expectation of a sum equals the sum of individual expectations
• Expressed as $E[aX + bY] = aE[X] + bE[Y]$ for constants a and b and random variables X and Y
• Holds true even when random variables are dependent
• Simplifies calculations involving complex combinations of random variables
• Applies to both discrete and continuous cases

Expectation of constants

• Expectation of a constant equals the constant itself: $E[c] = c$
• Allows for easy incorporation of fixed values in expectation calculations
• Useful when working with linear combinations of random variables and constants
• Simplifies expressions involving both random variables and deterministic values

Expectation of functions

• Calculates the average value of a function applied to a random variable
• Expressed as $E[g(X)] = \sum_{i=1}^{n} g(x_i) \cdot p(x_i)$ for discrete cases
• Continuous case formula: $E[g(X)] = \int_{-\infty}^{\infty} g(x) \cdot f(x) dx$
• Enables analysis of transformed random variables
• Useful for deriving moments and other statistical properties

Definition of variance

• Variance measures the spread or dispersion of a random variable around its expected value
• Plays a crucial role in Bayesian statistics for quantifying uncertainty and assessing the reliability of estimates

Measure of spread

• Quantifies the average squared deviation from the mean
• Expressed mathematically as $Var(X) = E[(X - \mu)^2]$, where μ is the expected value of X
• Provides insight into the variability and concentration of probability mass
• Larger variance indicates greater spread and more uncertainty
• Useful for comparing the dispersion of different probability distributions

Relationship to expectation

• Variance can be computed using expectations: $Var(X) = E[X^2] - (E[X])^2$
• Demonstrates the connection between second moment and first moment (mean)
• Allows for alternative calculation methods when direct computation is challenging
• Highlights the importance of both the average value and squared values in determining spread
• Provides a foundation for understanding higher-order moments and distribution shapes

Properties of variance

• Variance properties enable efficient analysis and manipulation of random variables in Bayesian statistics
• Facilitate the study of uncertainty propagation and error estimation in statistical models

Non-negativity

• Variance is always non-negative: $Var(X) \geq 0$
• Equals zero only for constants or degenerate random variables
• Reflects the fact that spread is measured as squared deviations
• Ensures consistency in interpreting variance across different distributions
• Provides a lower bound for uncertainty in statistical estimates

Variance of constants

• Variance of a constant is always zero: $Var(c) = 0$
• Indicates that constants have no uncertainty or variability
• Useful when working with combinations of random variables and fixed values
• Simplifies variance calculations for expressions involving constants

Variance of linear transformations

• For constants a and b, and random variable X: $Var(aX + b) = a^2 Var(X)$
• Demonstrates how scaling affects the spread of a distribution
• Shows that adding constants does not change the variance
• Enables analysis of how linear transformations impact uncertainty
• Useful in standardizing random variables and creating z-scores

Covariance and correlation

• Covariance and correlation measure the relationship between two random variables
• Essential concepts in Bayesian statistics for understanding dependencies and joint distributions

Definition of covariance

• Measures the joint variability of two random variables
• Expressed as $Cov(X,Y) = E[(X - E[X])(Y - E[Y])]$
• Positive covariance indicates variables tend to move together
• Negative covariance suggests inverse relationship
• Zero covariance implies no linear relationship (but does not rule out non-linear dependencies)

Correlation coefficient

• Normalized measure of linear dependence between two random variables
• Defined as $\rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$
• Ranges from -1 to 1, with -1 indicating perfect negative correlation and 1 perfect positive correlation
• Value of 0 suggests no linear correlation
• Unitless measure, allowing comparison of relationships between different variable pairs

Properties of correlation

• Symmetric: $\rho_{X,Y} = \rho_{Y,X}$
• Unchanged by linear transformations: $\rho_{aX+b,cY+d} = \rho_{X,Y}$ for constants a, b, c, d (a and c non-zero)
• Absolute value never exceeds 1: $|\rho_{X,Y}| \leq 1$
• Correlation of 1 or -1 implies perfect linear relationship
• Independent variables have zero correlation (but zero correlation does not imply independence)

Expectation in Bayesian inference

• Expectation plays a central role in Bayesian inference for parameter estimation and prediction
• Allows incorporation of prior knowledge and updating beliefs based on observed data

Prior expectation

• Represents the average value of a parameter before observing data
• Calculated using the prior distribution: $E_{\text{prior}}[\theta] = \int \theta p(\theta) d\theta$
• Encapsulates initial beliefs or expert knowledge about the parameter
• Serves as a starting point for Bayesian updating
• Influences posterior estimates, especially with limited data

Posterior expectation

• Average value of a parameter after incorporating observed data
• Computed using the posterior distribution: $E_{\text{posterior}}[\theta|x] = \int \theta p(\theta|x) d\theta$
• Combines prior knowledge with information from the likelihood
• Often used as a point estimate for the parameter of interest
• Represents updated beliefs about the parameter given the evidence

Predictive expectation

• Average value of future observations based on current knowledge
• Calculated using the predictive distribution: $E_{\text{pred}}[x_{\text{new}}|x] = \int x_{\text{new}} p(x_{\text{new}}|x) dx_{\text{new}}$
• Accounts for both parameter uncertainty and inherent randomness
• Useful for making predictions and assessing model performance
• Provides a single summary of expected future outcomes

Variance in Bayesian inference

• Variance quantifies uncertainty in Bayesian inference for parameters and predictions
• Crucial for assessing the reliability and precision of Bayesian estimates

Prior variance

• Measures the spread of the prior distribution for a parameter
• Calculated as $Var_{\text{prior}}(\theta) = E_{\text{prior}}[\theta^2] - (E_{\text{prior}}[\theta])^2$
• Reflects the initial uncertainty about the parameter before observing data
• Larger prior variance indicates less informative prior knowledge
• Influences the weight given to prior information in posterior calculations

Posterior variance

• Quantifies the remaining uncertainty about a parameter after observing data
• Computed using the posterior distribution: $Var_{\text{posterior}}(\theta|x) = E_{\text{posterior}}[\theta^2|x] - (E_{\text{posterior}}[\theta|x])^2$
• Generally smaller than prior variance due to information gained from data
• Used to construct credible intervals for parameter estimates
• Provides a measure of estimation precision in Bayesian analysis

Predictive variance

• Represents the uncertainty in future observations
• Calculated using the predictive distribution: $Var_{\text{pred}}(x_{\text{new}}|x) = E_{\text{pred}}[x_{\text{new}}^2|x] - (E_{\text{pred}}[x_{\text{new}}|x])^2$
• Accounts for both parameter uncertainty and inherent randomness in the data
• Useful for constructing prediction intervals
• Helps assess the reliability of model predictions

Moment-generating functions

• Moment-generating functions (MGFs) provide a powerful tool for analyzing probability distributions
• Play a significant role in Bayesian statistics for deriving distribution properties and making inferences

Definition and properties

• MGF of a random variable X defined as $M_X(t) = E[e^{tX}]$
• Exists in an open interval containing zero
• Uniquely determines the distribution if it exists
• MGF of sum of independent random variables is the product of their individual MGFs
• Useful for proving limit theorems and characterizing distributions

Relationship to expectation

• kth moment can be obtained by differentiating MGF k times and evaluating at t=0
• Expected value: $E[X] = M'_X(0)$
• Allows for easy computation of moments without direct integration
• Facilitates the derivation of expectations for transformed random variables
• Useful in Bayesian analysis for calculating expectations of complex functions

Relationship to variance

• Variance can be computed using the first and second derivatives of MGF
• $Var(X) = M''_X(0) - (M'_X(0))^2$
• Provides an alternative method for calculating variance
• Useful when direct computation of variance is challenging
• Enables analysis of how transformations affect the spread of distributions

Law of total expectation

• Fundamental theorem in probability theory with important applications in Bayesian statistics
• Allows decomposition of expectations based on conditional probabilities

Conditional expectation

• Expected value of a random variable given that another variable takes a specific value
• Denoted as $[E[Y|X]](https://www.fiveableKeyTerm:e[y|x])$, a function of the conditioning variable X
• Useful for analyzing relationships between variables in Bayesian models
• Provides insight into how one variable influences the average behavior of another
• Forms the basis for many Bayesian prediction and estimation techniques

Tower property

• States that $E[Y] = E[E[Y|X]]$
• Also known as the law of iterated expectations
• Allows computation of unconditional expectations using conditional expectations
• Simplifies complex expectation calculations by breaking them into steps
• Crucial in Bayesian inference for marginalizing over nuisance parameters

Law of total variance

• Extends the law of total expectation to variance calculations
• Important tool in Bayesian statistics for analyzing and decomposing uncertainty

Conditional variance

• Measures the variability of a random variable given the value of another variable
• Denoted as $Var(Y|X)$, a function of the conditioning variable X
• Quantifies the remaining uncertainty in Y after knowing X
• Useful for assessing the predictive power of one variable on another
• Often used in hierarchical Bayesian models to analyze multi-level variability

Decomposition of variance

• States that $Var(Y) = E[Var(Y|X)] + Var(E[Y|X])$
• Separates total variance into expected conditional variance and variance of conditional expectation
• First term represents average unexplained variance
• Second term quantifies variability explained by the conditioning variable
• Provides insight into sources of uncertainty in Bayesian models

Applications in Bayesian analysis

• Expectation and variance concepts form the foundation for various Bayesian analysis techniques
• Enable sophisticated statistical inference and decision-making under uncertainty

Parameter estimation

• Uses posterior expectation as point estimate for parameters
• Employs posterior variance to quantify estimation uncertainty
• Allows for incorporation of prior knowledge in the estimation process
• Facilitates the construction of credible intervals for parameters
• Enables comparison of different estimation methods (MAP, median, mean)

Hypothesis testing

• Utilizes Bayes factors to compare competing hypotheses
• Employs posterior probabilities to assess the plausibility of hypotheses
• Allows for continuous updating of beliefs as new evidence becomes available
• Provides a natural framework for model comparison and selection
• Enables decision-making based on expected losses or utilities

Decision theory

• Uses expected utility to guide optimal decision-making
• Incorporates both parameter uncertainty and consequences of actions
• Allows for formal treatment of risk and loss functions
• Facilitates the design of experiments to maximize information gain
• Enables adaptive strategies that update decisions as new data is observed

Computational methods

• Computational techniques play a crucial role in applying expectation and variance concepts in Bayesian statistics
• Enable analysis of complex models and high-dimensional problems

Monte Carlo estimation

• Approximates expectations using random sampling
• Estimates $E[g(X)]$ as $\frac{1}{n}\sum_{i=1}^n g(X_i)$ where X_i are samples from the distribution of X
• Allows for estimation of complex integrals and high-dimensional problems
• Provides unbiased estimates with quantifiable error bounds
• Forms the basis for many advanced Bayesian computation techniques (MCMC)

Importance sampling

• Improves Monte Carlo estimation efficiency for rare events or difficult-to-sample distributions
• Uses an alternative proposal distribution q(x) to estimate $E[g(X)] = \int g(x) \frac{p(x)}{q(x)} q(x) dx$
• Allows sampling from a simpler distribution while still estimating properties of the target distribution
• Reduces variance of estimates compared to naive Monte Carlo in many cases
• Crucial for estimating normalizing constants and marginal likelihoods in Bayesian models