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Related Concept Videos

Distributions to Estimate Population Parameter01:26

Distributions to Estimate Population Parameter

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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
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Confidence Intervals

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An unbiased point estimate is often insufficient to predict a population estimate, such as population mean or population proportion. In this scenario, a confidence interval is used. A confidence interval is an estimate similar to a sample proportion. However, unlike the point estimate which is a single value, the confidence interval contains a range of values. These values have lower and upper limits, known as confidence limits, and can be designated as L1 and L2, respectively.
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 The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
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Binomial Probability Distribution

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A binomial distribution is a probability distribution for a procedure with a fixed number of trials, where each trial can have only two outcomes.
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Posterior consistency in conditional distribution estimation.

Debdeep Pati1, David B Dunson2, Surya T Tokdar2

  • 1Department of Statistics, Florida State University.

Journal of Multivariate Analysis
|July 29, 2014
PubMed
Summary
This summary is machine-generated.

This study establishes conditions for accurate Bayesian estimation of conditional distributions using predictor-dependent Gaussian kernel mixtures. It ensures reliable posterior consistency for complex nonparametric models, advancing statistical inference.

Keywords:
AsymptoticsBayesian nonparametricsDensity regressionDependent Dirichlet processLarge supportProbit stick-breaking process

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Area of Science:

  • Statistics
  • Machine Learning
  • Nonparametric Bayesian Inference

Background:

  • Nonparametric Bayesian methods are crucial for estimating complex conditional distributions.
  • Ensuring posterior consistency is vital for reliable statistical inference.
  • Existing methods face challenges in estimating uncountable conditional distributions across predictor spaces.

Purpose of the Study:

  • To develop theoretical conditions for posterior consistency in nonparametric Bayesian estimation of conditional distributions.
  • To address the challenges of estimating distributions across diverse predictor regions.
  • To provide a framework for analyzing predictor-dependent priors.

Main Methods:

  • Defining topologies on the space of conditional distributions.
  • Formulating priors as predictor-dependent mixtures of Gaussian kernels.
  • Establishing sufficient conditions for posterior consistency.

Main Results:

  • Sufficient conditions for posterior consistency are provided for predictor-dependent Gaussian kernel mixtures.
  • These conditions are shown to be satisfied by generalized stick-breaking process mixtures with monotone, differentiable lengths.
  • Conditions are also derived for predictor-independent priors, including those from Dirichlet process priors.

Conclusions:

  • The developed theory offers a robust framework for nonparametric Bayesian conditional distribution estimation.
  • The findings are applicable to various prior constructions, enhancing the flexibility of Bayesian methods.
  • This work contributes to the theoretical understanding of posterior consistency in complex statistical models.