This is a collection of codes developed for Paulson et al. (2019).

Tools for performing the Bayesian analysis are contained within core_compute.py and plotting tools are in core_plot.py. Most of the remaining codes are scripts (e.g. Bayesian_Inference.py or liquid_lin.py) which perform a variety of Bayesian inference computations based on a common template. example_outlier.ipynb is a Jupyter notebook that demonstrates the use of the core_compute.py and core_plot.py for Bayesian inference.

For a quick primer to Bayesian statistics, see the Bayesian fundamentals - model calibration and selection notebook.

1. Run Bayesian_inference.py, changing the D['order'] parameter on line 87 to explore polynomials of varying order. This will produce Figures 9a, 10, and 11.
2. Run outliers_normal.py and outliers_students-t.py to produce Figures 12a and 12b, respectively.
3. Run errorbars_standard.py and errorbars_yma.py to produce Figures 13 and 14.
4. Run thermo_consistency.py and thermo_consistency_separate.py to produce Figure 1.

1. Run data_process/data_process_4.py
2. For each of alpha_quart_debye.py, beta_quad.py, and liquid_lin.py:
   • run a first time to get initial posterior
   • run a second time to use narrowed prior distributions and to evaluate the final marginal likelihoods
   • this will result in plots of the data/model-predictions with UQ, the univariate parameter distributions, a corner plot, a table with posterior statistics (used to produce Table 2), and a text output with the sampling time and marginal likelihood (used to produce Table 1)
3. Run plot_all.py to plot the data, model-predictions with UQ for each phase (alpha, beta, liquid)and property (Cp, H, S, G). This produces Figures 4 - 6.
4. Run plot_model_differences.py to plot the percent differences between the model prediction and previous Hf models (Figure 7).

• python/3.6.8
• emcee/2.2.1
• kombine/0.8.3
• matplotlib/3.0.2
• numpy/1.15.4
• pandas/0.23.4
• pymultinest/2.6
• scipy/1.2.1
• seaborn/0.9.0

Paulson, N.H., Jennings, E., Stan, M. “Bayesian strategies for uncertainty quantification of the thermodynamic properties of materials,” International Journal of Engineering Science. 142 (2019) 74-93 https://doi.org/10.1016/j.ijengsci.2019.05.011