Predictive Modeling and Bayesian Inference in R

Two applied statistical modeling exercises implemented from scratch in R using a reproducible RMarkdown workflow. The focus is on likelihood-based modeling, predictive evaluation, and simulation-based Bayesian inference.

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Part 1 — Predicting 3D Printed Object Weight

Modeled the relationship between CAD-estimated weight $x$ and actual weight $y$ using heteroscedastic Gaussian regression models.

Mathematical Framework

Each observation is modeled as:

$$ Y_i \sim \mathcal{N}(\mu_i, \sigma_i^2) $$

with mean structure

$$ \mu_i = \beta_1 + \beta_2 x_i $$

Two alternative variance structures were implemented.

Model A

$$ \sigma_i^2 = \exp(\beta_3 + \beta_4 x_i) $$

Model B

$$ \sigma_i^2 = \exp(\beta_3) + \exp(\beta_4) x_i^2 $$

Parameters were estimated via maximum likelihood by minimizing the negative log-likelihood.

Predictive Evaluation

For each observation, predictive distributions were constructed and evaluated using:

• Leave-one-out cross-validation
• Squared Error (point accuracy)
• Dawid–Sebastiani score (distributional accuracy)

A Monte Carlo test was used to assess whether one model provided significantly better predictive performance.

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Part 2 — Bayesian Estimation of Burial Counts

Estimated total population size $N$ and recovery probability $\phi$ from observed femur counts.

Model

$$ Y_1, Y_2 \sim \text{Binomial}(N, \phi) $$

with priors

$$ N \sim \text{Geometric}(\xi) $$

$$ \phi \sim \text{Beta}(a, b) $$

Posterior expectations were approximated using Monte Carlo integration:

$$ \hat{E}[N \mid y] = \frac{\sum_k N^{(k)} p(y \mid N^{(k)}, \phi^{(k)})} {\sum_k p(y \mid N^{(k)}, \phi^{(k)})} $$

All likelihood calculations were implemented using log-Gamma functions for numerical stability.

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Implementation Notes

• Custom log-likelihood functions
• Direct maximum likelihood estimation
• Predictive distribution construction
• Manual leave-one-out cross-validation
• Simulation-based Bayesian computation
• Fully reproducible RMarkdown workflow

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