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bayesics: Core Statistical Methods via Bayesian Inference in R

The R package **bayesics** offers a unified framework for Bayesian inference that simplifies statistical analysis by leveraging closed-form solutions and automated sampling to provide key inferential quantities and diagnostics without requiring users to master complex sampling algorithms.

Original authors: Daniel K. Sewell, Alan T. Arakkal

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel K. Sewell, Alan T. Arakkal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery. In the past, detectives (statisticians) had two main ways to work: the "Frequentist" way and the "Bayesian" way.

The Frequentist way is like a rigid rulebook. It tells you, "If you repeat this experiment a million times, 95% of the time you'll get a result like this." But it's notoriously bad at answering the question you actually care about: "Is my specific suspect guilty right now?" It often gives you confusing answers like, "We are 95% confident in the process we used," which doesn't tell you much about the specific case at hand. Plus, it relies heavily on "p-values," a metric that is often misunderstood and has led to many scientific mistakes.

The Bayesian way is more like a wise old detective who updates their beliefs as new evidence comes in. They start with a hunch (a prior), look at the evidence, and update their confidence. They can say, "I am 95% sure the suspect is guilty," which is exactly what you want to hear.

However, there's a catch: doing Bayesian math is usually like trying to bake a soufflé while juggling chainsaws. It requires complex computer algorithms (sampling) that are hard to tune, easy to mess up, and require a PhD in computer science to get right. If you don't get the algorithm perfect, your delicious-looking soufflé (your results) might actually be raw dough.

Enter `bayesics`: The "Automatic Chef"

The paper introduces a new R software package called bayesics. Think of bayesics as an automatic chef that lets you order a gourmet Bayesian meal without needing to know how to cook, how to tune the oven, or how to juggle chainsaws.

Here is how it works, using simple analogies:

1. The "No-Touch" Kitchen (Closed-Form Solutions)

Most Bayesian tools force you to manually stir the pot (run complex sampling algorithms) for hours. bayesics is different. Whenever possible, it uses closed-form solutions.

The Analogy: Instead of making a sauce from scratch by chopping, sautéing, and simmering for hours, bayesics has a secret stash of "magic instant packets." If the math allows, it just mixes the packet with water, and poof—you have the perfect sauce instantly. You don't need to know the chemistry; you just get the result.

2. The "Self-Adjusting" GPS (Automatic Sampling)

Sometimes, the "magic packet" doesn't exist, and you do have to cook from scratch (use sampling). In other software, you have to guess how long to cook it. If you cook it too little, it's raw; too much, it's burnt.

The Analogy: bayesics has a built-in GPS. It takes a quick "taste test" (a small sample) to see how complex the dish is. If the sauce is thick and hard to mix, the GPS says, "Okay, we need 10,000 more stirs to get this right." If it's thin, it says, "500 stirs is enough." It automatically decides exactly how much computing power you need to get a precise answer, so you never have to guess.

3. The "Truth-Teller" Dashboard (Inference over Algorithms)

Many tools show you a dashboard full of technical warnings about the engine (the computer algorithm): "Are the wheels spinning? Is the engine hot?"

The Analogy: bayesics ignores the engine noise. It only shows you the dashboard of the car's destination. It gives you the answers that actually matter:
- The Best Guess: "Where do we think the truth is?"
- The Safety Net (Credible Intervals): "We are 95% sure the truth is between these two numbers." (Unlike Frequentist confidence intervals, which are often misinterpreted, this one actually means what it says).
- The Direction: "Is the relationship positive or negative?"
- The "Real World" Check (ROPE): "Is this difference big enough to actually matter, or is it just a tiny, meaningless fluctuation?"
- The Evidence Score (Bayes Factors): "How much more likely is my theory than the opposite theory?"

4. The "Quality Control" Inspector (Diagnostics)

Even with a great chef, sometimes the ingredients are bad. If your data doesn't fit the model, you need to know.

The Analogy: bayesics has a quality control inspector. It runs a "Bayesian P-value" test. Imagine you are trying to fit a square peg into a round hole. The inspector will scream, "Hey! This shape doesn't fit!" If the model is wrong, bayesics doesn't just give you a bad result; it offers non-parametric alternatives (like a flexible rubber peg) that can bend to fit the data without breaking.

5. The "Mediation" Fixer

The paper also mentions that bayesics fixes some broken tools used for "Mediation Analysis" (figuring out how one thing causes another).

The Analogy: Other packages were like a map with a missing bridge. bayesics builds the bridge, ensuring the path from Cause to Effect is mathematically sound and accurate.

Why Should You Care?

The authors argue that science is currently stuck in a loop: "We teach students Frequentist statistics because that's what journals want, and journals want it because that's what they were taught."

bayesics wants to break that loop. It makes the superior Bayesian method accessible to everyone. You don't need to be a math wizard or a coding expert. You just need to ask the question, and bayesics will handle the heavy lifting, the cooking, and the quality control, handing you a clear, honest, and interpretable answer.

In short: bayesics is the tool that finally lets you drive a high-performance Bayesian car without needing a mechanic's license.

1. Problem Statement

The paper addresses several critical gaps in the current landscape of Bayesian statistical software in R:

Algorithmic Complexity vs. Inference Needs: Many existing Bayesian packages (e.g., rstanarm, brms, NIMBLE) require users to understand complex sampling algorithms (MCMC), convergence diagnostics (e.g., $\hat{R}$ , Geweke), and tuning parameters. This creates a barrier to entry for applied scientists who need reliable inference but lack deep algorithmic expertise.
Misleading Accuracy Metrics: Standard diagnostics like the Monte Carlo Standard Error (MCSE) are often used to assess the accuracy of posterior estimates. However, the authors demonstrate that MCSE is reliable only for point estimates (means) and can be highly misleading for interval estimates (credible intervals). Obtaining sufficient precision for tail quantiles (interval bounds) often requires orders of magnitude more samples than for point estimates, a nuance often ignored by default settings in other software.
Fragmented Functionality: Existing packages often specialize in specific tasks (e.g., BayesFactor for hypothesis testing, arm for regression) or require building models from scratch. There is a lack of a unified framework that provides a consistent syntax for common procedures (one/two-sample tests, regression, mediation) while automatically handling the computational burden.
Interpretability: Frequentist outputs (p-values, confidence intervals) are frequently misinterpreted. The paper argues for a framework that prioritizes directly interpretable Bayesian quantities (e.g., probability of direction, ROPE) over algorithmic details.

2. Methodology

The bayesics package is designed around the philosophy of "Inference, not Algorithms." It achieves this through three primary methodological strategies:

A. Automated Sample Size Determination

Instead of relying on fixed sample sizes or user-defined convergence criteria, bayesics automatically calculates the number of posterior samples required to achieve a specific precision for interval estimates.

Theoretical Basis: It utilizes a Central Limit Theorem for order statistics (Doss et al., 2014) to determine the sample size ( $L$ ) needed to estimate a quantile (e.g., the lower bound of a 95% CI) within a margin of error $\epsilon$ with probability $s$ .
Implementation: For models without conjugate priors, the package draws a preliminary sample to estimate the posterior density at the desired quantile, then calculates and draws the remaining samples necessary for the target precision.

B. Hybrid Computational Approaches

Closed-Form Solutions: Where possible, the package uses conjugate priors to derive exact posterior distributions, eliminating the need for sampling entirely.
- Example: lm_b uses a Normal-Inverse Gamma prior for linear regression, yielding a multivariate t-distribution for coefficients.
Variational Bayes (VB) & Approximations: For Generalized Linear Models (GLMs) where conjugate priors do not exist, bayesics implements fixed-form Variational Bayes (minimizing Kullback-Leibler divergence) as the default, with options for importance sampling and large-sample approximations.
Non-Parametric Methods: When parametric assumptions fail, the package offers:
- Loss-Likelihood Bootstrap: A non-parametric approach using general Bayesian updating (Lyddon et al., 2019) that avoids specifying priors on parameters or assuming a correct likelihood function.
- Semi-Parametric Survival Analysis: Implements a piecewise exponential hazard model with closed-form posteriors for survival curves.

C. Unified Inferential Framework

The package standardizes the output across all functions to provide a consistent set of inferential quantities:

Point Estimates: Posterior mean, median, or mode.
Interval Estimates: Credible Intervals (CIs).
Probability of Direction (PDir): The probability that a parameter is positive or negative.
Region of Practical Equivalence (ROPE): The probability that the parameter lies within a range of practical insignificance.
Bayes Factors: For model comparison and hypothesis testing.
Information Criteria: AIC, BIC, DIC, and WAIC.

3. Key Contributions

Unified Syntax and Output: bayesics provides a single interface for a wide range of analyses (ANOVA, t-tests, regression, mediation, survival analysis) with consistent generic functions (coef, credint, plot, summary, bayes_factors).
Correction of Existing Implementations:
- Mediation Analysis: Provides a fully Bayesian implementation of the Imai et al. (2010) causal mediation framework, correcting the "quasi-Bayesian" approach (improper priors) used in the popular mediation package.
- ROPE Bounds: Addresses issues in bayestestR regarding default ROPE bounds by adjusting them according to the scale of covariates.
Diagnostic Tools:
- Bayesian p-values: Implements posterior predictive p-values to assess model fit (compatibility of model features with observed data).
- Visual Diagnostics: Provides residual plots, QQ-plots, and scatterplots of test statistics for model assessment.
Robustness: Offers non-parametric alternatives (e.g., np_glm_b) that automatically widen credible intervals when model assumptions are violated, preventing overconfident inferences.

4. Results and Demonstration

The paper demonstrates the package using a randomized clinical trial dataset (indomethacin vs. placebo for preventing post-ERCP pancreatitis):

Randomization Checks: Successfully used t_test_b, prop_test_b, and wilcoxon_test_b to verify covariate balance, providing Bayes factors and credible intervals for differences.
Model Diagnostics: Used Bayesian p-values (based on deviance) to validate a GLM. A p-value near 0.5 indicated a good fit, whereas a value near 0 or 1 would have triggered a switch to the non-parametric np_glm_b.
Inference: The analysis of the treatment effect (rx) showed:
- A 52.8% reduction in the odds of PEP (95% CI: 22.7% - 71.2%).
- A Probability of Direction (PDir) of 0.999.
- A Bayes Factor of 10.6 (Strong evidence) favoring the treatment effect.
- A negligible probability (0.006) that the effect falls within the ROPE, confirming practical significance.
Visualization: Generated credible bands for covariates, automatically fixing other variables at their medoid (most central) observation to ensure realistic extrapolation.

5. Significance

The bayesics package represents a significant step toward democratizing Bayesian inference in applied science.

Accessibility: By abstracting away algorithmic tuning and sampling diagnostics, it allows researchers to focus on scientific interpretation rather than computational mechanics.
Reliability: It directly addresses the "false sense of accuracy" problem by ensuring that interval estimates are computed with sufficient precision, a critical flaw in many default settings of other MCMC-based packages.
Educational Value: It serves as a pedagogical tool, demonstrating how to correctly interpret Bayesian quantities (PDir, ROPE, Bayes Factors) in a unified manner.
Future-Proofing: By integrating corrections to existing methods (mediation) and offering non-parametric fallbacks, it provides a robust framework for modern statistical practice where model assumptions are often uncertain.

In summary, bayesics bridges the gap between theoretical Bayesian rigor and practical application, offering a "basics" framework that is both statistically sound and user-friendly.

Enter bayesics: The "Automatic Chef"