To tackle Exercise $11\mathrm{.}13$ from your assignment, which focuses on comparing four different models of penguin body mass using Bayesian methods, we'll proceed step$-$by$-$step as outlined:

Exercise $11\mathrm{.}13$: Penguins! Comparing Models

a$.$ Simulate Models:

First, we need to simulate the four models based on the provided formulas:

Model $1$: body$\_$mass$\_$g ~ flipper$\_$length$\_$mm

Model $2$: body$\_$mass$\_$g ~ species

Model $3$: body$\_$mass$\_$g ~ flipper$\_$length$\_$mm $+$ species

Model $4$: body$\_$mass$\_$g ~ flipper$\_$length$\_$mm $+$ bill$\_$length$\_$mm $+$ bill$\_$depth$\_$mm

b$.$ Produce and Compare Plots:

After simulating the models, we'll produce plots to visually compare them. These plots will likely include:

Explanation:

Scatter plots of observed vs$.$ predicted body mass.

Posterior distributions of parameters for each model.

Any other relevant diagnostic plots $($like residual plots if needed$).$

c$.$ Use $10-$fold Cross$-$validation:

We'll use $10-$fold cross$-$validation to assess and compare the posterior predictive quality of the models. This involves:Implementing cross$-$validation using prediction$\_$summary$\_$cv$()$ or similar functions.

Explanation:Implementing cross$-$validation using prediction$\_$summary$\_$cv$()$ or similar functions.

Comparing metrics like mean squared error $($MSE$),$ root mean squared error $($RMSE$),$ or others as appropriate.

d$.$ Evaluate and Compare ELPD:

We will evaluate and compare the expected log predictive density $($ELPD$)$ to assess the predictive accuracy of each model. This step is crucial in Bayesian model comparison to understand which model performs better in predicting unseen data.

e$.$ Summary and Conclusion:

Finally, we'll summarize the findings and discuss which of the four models appears to be the "best" based on the evaluations in steps c and d$.$

Detailed Approach

Explanation:

Simulate Models: Define each model using the specified formulas and simulate data based on these models. Ensure the simulation captures the essence of the relationships described $($e$.$g$.,$ linear relationships, species$-$specific effects$).$

Produce and Compare Plots: Create plots to compare observed vs$.$ predicted body mass across the models. Also, visualize posterior distributions of model parameters to understand uncertainties and differences between models.

Cross$-$validation: Implement $10-$fold cross$-$validation to obtain metrics that quantify each model's predictive performance. This step ensures robust evaluation by simulating how each model generalizes to new data.

ELPD Comparison: Compute and compare ELPD values across models to determine which model provides the best balance between fit and complexity. Lower ELPD indicates better predictive performance.

Conclusion: Based on the results from cross$-$validation and ELPD comparison, conclude which model is preferred and why. Discuss the strengths and weaknesses of each model in light of the penguin body mass prediction task.

Submission Format

Your submission should include:

Explanation:

An HTML document containing:

All R code used for simulations, plots, cross$-$validation, and ELPD calculations.

Explanations of each step, including interpretations of results.

Outputs such as plots and tables.

An $.$Rmd file $($or extracted from the HTML$)$ with the same content, for reproducibility.

Ensure the document is well$-$organized with appropriate headers and sections corresponding to each part of the exercise. References to relevant course materials or textbooks should support your interpretations and choices throughout the analysis.

By following this structured approach, you can systematically compare the models and provide a clear, well$-$supported conclusion on which model is best suited for predicting penguin body mass based on the given data.

AS PER ABOVE SITUATION, WRITE THE CODE.

NEED CODE FOR