feat: new article

Author: DiogoRibeiro7

_posts/2024-11-15-a critical examination of bayesian posteriors as test statistics.md

@@ -74,6 +74,7 @@ where:
 The posterior combines the prior information with the likelihood, producing a probability distribution over $$\theta$$ that reflects both prior beliefs and observed data.
 
 ### Comparing Likelihoods and Posteriors
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 While both the likelihood function and the posterior distribution involve $$p(x \mid \theta)$$, they serve different purposes:
 
 - **Likelihood Function:** Used in frequentist inference for parameter estimation and hypothesis testing, focusing on the data's information about $$\theta$$.
@@ -82,16 +83,19 @@ While both the likelihood function and the posterior distribution involve $$p(x
 When the prior $$p(\theta)$$ is non-informative or uniform, the posterior is proportional to the likelihood. This similarity has led some to argue that the posterior, in such cases, acts merely as a scaled version of the likelihood function.
 
 ### Interpretation and Misinterpretation
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 A key point of contention arises in interpreting the posterior distribution as a probability distribution over parameters. In frequentist statistics, parameters are fixed but unknown quantities, and probabilities are associated only with data or statistics derived from data. In contrast, Bayesian statistics treat parameters as random variables, allowing for probability statements about them.
 
 Critics argue that when the posterior is viewed as a test statistic, especially in cases with non-informative priors, interpreting the area under its tail or its ratios as probabilities can be misleading. They contend that without meaningful prior information, the posterior does not provide genuine probabilistic evidence about $$\theta$$ but rather serves as a transformed version of the likelihood.
 
 ## Test Statistics and Their Role in Statistical Inference
 
 ### Definition of Test Statistics
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 A test statistic is a function of the sample data used in statistical hypothesis testing. It summarizes the data into a single value that can be compared against a theoretical distribution to determine the plausibility of a hypothesis. The choice of test statistic depends on the hypothesis being tested and the underlying statistical model.
 
 ### Properties of Good Test Statistics
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 An effective test statistic should have the following properties:
 
 - **Sufficiency:** Captures all the information in the data relevant to the parameter of interest.
@@ -100,6 +104,7 @@ An effective test statistic should have the following properties:
 - **Robustness:** Performs well under various conditions, including deviations from model assumptions.
 
 ### Sufficient Statistics
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 A sufficient statistic is a function of the data that contains all the information needed to estimate a parameter. Formally, a statistic $$T(x)$$ is sufficient for parameter $$\theta$$ if the conditional distribution of the data $$x$$ given $$T(x)$$ does not depend on $$\theta$$:
 
 $$
@@ -109,18 +114,21 @@ $$
 Sufficient statistics are valuable because they reduce data complexity without losing information about the parameter. They play a crucial role in both estimation and hypothesis testing.
 
 ### Role in Decision-Making
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 In hypothesis testing, the decision to reject or fail to reject the null hypothesis is based on the test statistic's value relative to a critical value or significance level. The test statistic's distribution under the null hypothesis determines the probabilities associated with different outcomes.
 
 Critics argue that the long-run performance of a test statistic, driven by the sufficient statistic, is what ultimately matters in statistical inference. Scaling or transforming a test statistic does not change its essential properties or its ability to make accurate decisions in the long run.
 
 ## Scaling and Normalization of Likelihoods
 
 ### Impact of Scaling on Test Statistics
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 Scaling and rescaling a test statistic involve multiplying or transforming it by a constant or function. While such transformations can change the numerical values of the statistic, they do not alter its fundamental properties or its distribution under repeated sampling.
 
 For example, if $$Z$$ is a test statistic, then $$c \cdot Z$$ (where $$c$$ is a constant) is a scaled version of $$Z$$. The scaling factor $$c$$ can adjust the magnitude but does not affect the statistic's ability to distinguish between hypotheses.
 
 ### Long-Run Performance
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 The long-run performance of a test statistic refers to its behavior over many repetitions of an experiment. Key considerations include:
 
 - **Type I Error Rate:** The probability of incorrectly rejecting the null hypothesis when it is true.
@@ -130,18 +138,21 @@ The long-run performance of a test statistic refers to its behavior over many re
 These properties are inherent to the test statistic's distribution and are not affected by scaling or normalization. Therefore, the focus should be on the statistic's ability to make accurate decisions rather than its scaled values.
 
 ### Importance of Sufficient Statistics
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 Since sufficient statistics capture all relevant information about the parameter, they determine the test statistic's long-run performance. Any transformation that retains sufficiency will preserve the statistic's essential properties.
 
 Scaling and rescaling may be employed for convenience or interpretability but do not enhance the test statistic's efficacy. Consequently, excessive manipulation of the likelihood or posterior may be unnecessary if it does not contribute to better inference.
 
 ## Appropriate Lexicon and Notation in Presenting Likelihoods
 
 ### Misuse of Bayesian Terminology
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 Presenting scaled likelihoods or transformed test statistics using Bayesian lexicon and notation, such as invoking Bayes' theorem, can be misleading. This practice may suggest that the resulting quantities are probabilities when they are not.
 
 For instance, integrating a scaled likelihood over a parameter space and interpreting the area as a probability disregards the fact that the likelihood function is not a probability distribution over parameters. Unlike probability densities, likelihoods do not necessarily integrate to one and can take on values greater than one.
 
 ### Need for Clarity and Precision
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 Using appropriate terminology and notation is crucial for clear communication in statistical analysis. Misrepresenting likelihoods as probabilities can lead to incorrect interpretations and conclusions.
 
 Practitioners should:
@@ -151,18 +162,21 @@ Practitioners should:
 - **Provide Context:** Explain the meaning and purpose of scaled or normalized quantities to prevent misunderstandings.
 
 ### Emphasizing the Nature of the Likelihood
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 By presenting the likelihood function in its proper context, analysts can avoid overstating its implications. Recognizing that the area under a likelihood curve is not a probability helps maintain the distinction between likelihood-based inference and probabilistic statements about parameters.
 
 ## Challenges with Scaled, Normalized, and Integrated Likelihoods
 
 ### Difficulty in Obtaining Standard Distributions
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 When likelihoods are scaled, normalized, or integrated, the resulting quantities may not follow standard statistical distributions. This lack of standardization presents challenges:
 
 - **Non-Standard Distributions:** The transformed likelihood may not conform to well-known distributions like the normal, chi-squared, or t-distributions.
 - **Complexity in Inference:** Without a standard distribution, it becomes difficult to calculate critical values, p-values, or confidence intervals.
 - **Analytical Intractability:** The mathematical expressions may be too complex to handle analytically, requiring numerical methods.
 
 ### Need for Transformations or Simulations
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 To make use of scaled or integrated likelihoods, further steps are often necessary:
 
 - **Transformation to Known Distributions:** Applying mathematical transformations to map the likelihood to a standard distribution.
@@ -171,6 +185,7 @@ To make use of scaled or integrated likelihoods, further steps are often necessa
 These additional steps add complexity to the analysis and may not provide sufficient benefits to justify their use.
 
 ### Questioning the Practical Utility
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 Given the challenges associated with scaled and normalized likelihoods, one may question their practicality:
 
 - **Added Complexity Without Clear Benefit:** The effort required to manipulate the likelihood may not yield better inference or understanding.
@@ -182,16 +197,19 @@ The critical view suggests that using intractable test statistics complicates th
 ## The Critique of Bayesian Probability Interpretations
 
 ### Over-Interpretation of Bayesian Posteriors
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 Some critics argue that Bayesian practitioners may overstate the implications of posterior distributions by treating them as definitive probabilities about parameters. This perspective contends that without meaningful prior information, the posterior is merely a transformed likelihood and does not provide genuine probabilistic evidence.
 
 The concern is that the probabilistic interpretation of the posterior may be unwarranted, especially when the prior is non-informative or subjective.
 
 ### Reliance on Sufficient Statistics
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 From a frequentist standpoint, the decision to retain or reject a hypothesis should rely on sufficient statistics derived from the data. The focus is on the long-run frequency properties of the test statistic, which are determined by the sufficient statistic.
 
 The argument is that introducing Bayesian probabilities does not enhance the decision-making process if the sufficient statistic already captures all relevant information.
 
 ### Implications for Hypothesis Testing
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 The critique extends to the practical application of Bayesian methods in hypothesis testing:
 
 - **Evidence vs. Decision:** Bayesian posteriors provide a probability distribution over parameters but may not directly inform the decision to accept or reject a hypothesis.
@@ -201,20 +219,23 @@ The critique extends to the practical application of Bayesian methods in hypothe
 ### Rebuttals and Counterarguments
 
 #### Defense of Bayesian Methods
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 Proponents of Bayesian statistics offer several counterarguments:
 
 - **Probabilistic Interpretation:** Bayesian methods provide a coherent probabilistic framework for inference, allowing for direct probability statements about parameters.
 - **Incorporation of Prior Information:** The ability to include prior knowledge can enhance inference, especially in cases with limited data.
 - **Flexibility and Adaptability:** Bayesian approaches can handle complex models and hierarchical structures more readily than frequentist methods.
 
 #### Value in Decision-Making
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 Bayesian posteriors can inform decision-making through:
 
 - **Credible Intervals:** Providing intervals within which the parameter lies with a certain probability.
 - **Bayes Factors:** Offering a method for model comparison and hypothesis testing based on the ratio of marginal likelihoods.
 - **Decision-Theoretic Framework:** Facilitating decision-making by incorporating loss functions and expected utility.
 
 #### Addressing the Critique
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 - **Objective Priors:** Using objective or reference priors to minimize subjectivity.
 - **Emphasis on Posterior Predictive Checks:** Assessing model fit and predictive performance rather than relying solely on the posterior distribution.
 - **Recognition of Limitations:** Acknowledging the challenges and working towards methods that address concerns about interpretation and practicality.