Methods Flashcards
Bayes’ Rule
Bayes’ Rule allows for the updating of prior beliefs about parameters (θ) in light of new evidence (y). The posterior distribution, which represents our updated beliefs, is given by: p(θ|y) is proportional to the product of the prior p(θ) and the likelihood p(y|θ). This rule is foundational in Bayesian analysis as it provides a systematic way to revise beliefs.
Marginal Likelihood Calculation
Marginal Likelihood Calculation involves computing the total likelihood of the observed data across all possible values of the parameters. This is represented as p(y) = ∫ p(θ)p(y|θ)dθ. This integral can be complex, often requiring numerical methods for evaluation, and is crucial for model comparison and selection.
Bayesian Inference Techniques
Bayesian Inference Techniques utilize the posterior distribution p(θ|y) to derive estimates of the parameters. Common techniques include calculating the posterior mean, which provides a point estimate, and credible intervals that reflect uncertainty. The inference allows one to quantify the uncertainty in estimates and make probabilistic statements about the parameters.
Savage-Dickey Density Ratio
The Savage-Dickey Density Ratio is a powerful tool for hypothesis testing in Bayesian statistics. It provides a method to compare nested models by evaluating the ratio of posterior densities at a specific parameter value under the null hypothesis. This ratio is computed as BF = p(θ0|y) / p(θ0|H0), where θ0 represents the hypothesized value and H0 is the null model.
Monte Carlo Integration
Monte Carlo Integration is a technique used to estimate expected values by generating random samples from a distribution. Specifically, to estimate E[h(θ)|y], one draws M samples θ(m) from the posterior p(θ|y) and computes the average: E[h(θ)|y] ≈ (1/M) Σ h(θ(m)). This method is particularly useful in high-dimensional spaces where analytical integration is infeasible.
Markov Chain Monte Carlo (MCMC)
Markov Chain Monte Carlo (MCMC) methods are designed to generate samples from complex posterior distributions by constructing a Markov chain. The samples generated by the chain will converge to the desired distribution, enabling the estimation of posterior characteristics. Common MCMC algorithms include Gibbs sampling and the Metropolis-Hastings algorithm.
Importance Sampling
Importance Sampling is a statistical technique used to estimate properties of a distribution by sampling from a different, more manageable distribution. It involves computing weighted averages using a proposal distribution g(θ) that approximates the target distribution p(θ|y). The weights are defined as w(θ) = p(θ|y) / g(θ), ensuring that the samples are adjusted to accurately represent the target.
Gibbs Sampling
Gibbs Sampling is an iterative method for obtaining samples from the joint distribution of multiple variables by sampling each variable conditionally on the others. In each iteration, a variable is updated by sampling from its conditional distribution given the current values of all other variables. This method is particularly effective for high-dimensional models.
Metropolis-Hastings
The Metropolis-Hastings algorithm generates samples from a target distribution by proposing new samples and accepting or rejecting them based on a calculated acceptance probability. The algorithm ensures that the resulting sample is distributed according to the target distribution. This flexibility allows it to sample from distributions that are difficult to sample from directly.
Slice Sampler
The Slice Sampler works by defining horizontal slices of the target distribution and uniformly sampling from these slices. This technique allows for efficient sampling from complex distributions without requiring knowledge of the normalizing constant, making it particularly useful for high-dimensional settings.
Hamiltonian Monte Carlo
Hamiltonian Monte Carlo (HMC) combines concepts from physics with statistical sampling. By treating parameters as particles moving through a potential energy landscape defined by the posterior distribution, HMC uses gradients to guide the sampling process, resulting in more efficient exploration of complex parameter spaces compared to random-walk proposals.
Data Augmentation
Data Augmentation involves introducing latent or unobserved variables into the model to simplify the estimation of the posterior distribution. By augmenting the data with these latent variables, one can leverage existing sampling techniques to sample from the joint distribution of observed and latent variables, improving convergence properties.
Bayesian Variable Selection
Bayesian Variable Selection employs prior distributions that encourage sparsity among the model coefficients. This approach allows for the identification of relevant predictors while automatically penalizing less informative ones. Common priors used include the Laplace or horseshoe priors, which facilitate the selection of variables based on posterior inclusion probabilities.
Analytical Methods for Multivariate Regression
Analytical Methods for Multivariate Regression provide closed-form solutions for the posterior distributions when dealing with multiple dependent variables. By leveraging the properties of the multivariate normal distribution, one can derive the joint posterior distributions of the regression coefficients and error covariance structures efficiently.
Covariance Structures
Covariance Structures describe the relationships between multiple random variables within a model. Defining the covariance matrix Σ is crucial for capturing the dependencies between errors in multivariate regression models. The structure can be specified based on prior knowledge or empirical evidence, guiding the estimation process.
Griddy Gibbs Sampler
The Griddy Gibbs Sampler is a modification of Gibbs sampling that discretizes the parameter space, which can improve sampling efficiency, especially in high-dimensional models. By defining a grid over the parameter space, the sampler draws samples from discrete points, facilitating convergence in cases where standard Gibbs sampling may struggle.
Hierarchical Bayes
Hierarchical Bayes approaches involve specifying prior distributions that depend on hyperparameters, allowing for more flexible modeling of complex data structures. By introducing these levels of priors, one can capture additional sources of variability and improve the estimation of parameters across different groups.
Panel Data Models
Panel Data Models analyze data that includes multiple observations over time for the same subjects. These models enable the estimation of both time-invariant and time-varying effects, offering insights into dynamic behaviors across individuals, firms, or countries.
Stochastic Multi-Level Models
Stochastic Multi-Level Models incorporate random effects at multiple levels to account for unobserved heterogeneity in the data. This approach allows for modeling the variability within and between groups, improving the accuracy of predictions and inferences.
Random Effects Models
Random Effects Models assume that individual-specific effects are not correlated with the explanatory variables, allowing for efficient estimation of parameters when dealing with panel data. This modeling approach can account for the unobserved variability across different individuals while estimating the average effects of covariates.
Bayesian Model Selection
Bayesian Model Selection uses posterior probabilities or Bayes factors to evaluate competing models based on the observed data. By calculating the probability of the data under different models, researchers can choose the model that best explains the data while taking into account model complexity.
