Prior and Posterior Probability
Prior and posterior probabilities are essential concepts in Bayesian inference, providing a way to update our beliefs about uncertain parameters based on observed data.
Prior Probability
- Prior probability represents our initial belief about the parameters before observing any data.
- If [Tex]\theta[/Tex]is the parameter of interest, the prior probability distribution[Tex]P(\theta) [/Tex] captures our uncertainty about [Tex]\theta [/Tex] before seeing the data.
- It reflects our existing knowledge, expert opinions, or assumptions about the parameter values.
- The prior guides the analysis by influencing the posterior distribution, which represents updated beliefs after observing the data.
Posterior Probability
- Posterior probability represents our updated belief about the parameters after incorporating data.
- It is computed using Bayes’ theorem, which combines the likelihood of the data given the parameters and the prior probability of the parameters to compute the posterior probability distribution.
- If D is the observed data, the posterior probability distribution[Tex]P(\theta|D)[/Tex] is:
- [Tex]P(\theta|D) = \frac{P(D|\theta) \times P(\theta)}{P(D)} [/Tex]
- The posterior distribution captures our uncertainty about the parameters given the observed data.
Bayesian Model Selection
Bayesian Model Selection is an essential statistical method used in the selection of models for data analysis. Rooted in Bayesian statistics, this approach evaluates a set of statistical models to identify the one that best fits the data according to Bayesian principles. The approach is characterized by its use of probability distributions rather than point estimates, providing a robust framework for dealing with uncertainty in model selection.
Table of Content
- What is the Bayesian Model Selection?
- Bayesian Inference
- Key Components of Bayesian Statistics
- Prior and Posterior Probability
- Prior Probability
- Posterior Probability
- Model Comparison Techniques
- Bayesian Factor (BF)
- Bayesian Information Criterion (BIC)
- Advantages of Bayesian Model Selection
- Conclusion
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