PSY3213 Introduction to Bayesian Data Analysis

Page 1

Introduction

• Outline of Topics:
  • I. Some Motivation
  • II. The Fundamental Framework for Bayesian Analysis
  • III. Some Simple Examples
  • IV. The $64,000 Question
  • V. Open Discussion

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Hurricane Prediction Questions

• Key Questions:
  • Where is the hurricane going to go?
  • When will it make landfall?
  • How severe will it be upon landfall?

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Factors Influencing Hurricane Path

• Influential Factors:
  • Other storm systems
  • Surface water temperature
  • Traveling over land

Weather Analysis Tools

• Mention of U.S. Surface Analysis with Radar and IR Satellite data
• Updated data presented, including wind pressure and tropical storm conditions.

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Factors Affecting Hurricane Paths

• Surface water temperatures significantly impact hurricanes.
• Traveling over land lessens storm strength.

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The Cone of Uncertainty

• Model Predictions:
  • In 1990, the average three-day forecast error was 300 nautical miles.
  • By 2020, it reduced to 100 miles, with an average 8-mile error for final model and actual landfall prediction.
  • The accuracy of five-day forecasts improved to match that of three-day forecasts from 2001.

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Advantages of Bayesian Data Analysis

• Comparison to NHST (Null Hypothesis Significance Testing):
  • NHST determines how likely data occurs by chance under a null hypothesis.
  • Bayesian analysis evaluates how likely the hypothesis is given the observed data, emphasizing the support for either hypothesis.

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Practical Benefits of Bayesian Analysis

• Null effects provide equivalent usefulness to rejecting the null hypothesis.
• Bayes supports assessing all models relevant to evidence without the need for one hypothesis to overshadow the other.

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Iterative Testing in Bayesian Analysis

• Bayesian methods allow iterative analysis, reviewing data to decide if more data is needed.
• Contrast with NHST, where sample sizes must be fixed in advance, creating constraints during experimentation.

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Robustness of Bayesian Conclusions

• Conclusions are less susceptible to incorrect results from small, unrepresentative sample sizes.
• Bayesian methods do not excessively rely on minority subjects, reducing bias from exclusion decisions.

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Small Sample Validity

• Bayesian analyses maintain validity irrespective of sample size, opposing the NHST reliance on asymptotic results.

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Steps in Bayesian Analysis

Step 1: Establish Prior

• Set initial beliefs about parameters based on previous knowledge.

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Step 2: Compute Likelihood

• Calculate the likelihood of collected data for each different model being considered.

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Step 3: Calculate Posterior Probability

• Posterior probability for models is obtained by multiplying priors and likelihoods, with assistance from computational tools when necessary.

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Coin Example in Bayesian Analysis

• Assessing if a coin is fair involves defining possible models ($ heta1 = 0.25, heta2 = 0.5, heta_3 = 0.75$).

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Likelihood Calculation for Coin Toss

• Likelihood based on data showing heads/tails outcomes (12 flips: 3 heads, 9 tails).
• Likelihood formula: $p(D| heta) = heta^H(1- heta)^{N-H}$
• Example parameters lead to assessing the likelihood for fair versus biased coins.

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Posterior Calculation

• Posterior computation requires normalization constant adjustment—from likelihood and priors