What are the most important tools for visualizing data in Bayesian statistics and decision analysis?

I usually use matplotlib to plot the histograms and scipy to fit the distribution. Then, I plot the fitted function on the same plot to see the quality of the fitting. Seaborn could be a good alternative to matplotlib.

I choose Bayesian analysis because I prefer to think about the data generating process explicitly. As such, using a histogram to visualize the data can support (or contradict) my assumptions of the likelihood distribution. For example, if my data are asymmetric, a Gaussian (normal) likelihood is probably wrong. If I see lots of outliers, I might consider that my data are generated by two processes, indicating a mixture of normals which leads to a joint distribution with thicker tails than a Gaussian. For this, I almost always use a histogram, with enough bins to give a general idea of the distribution's shape.

Posterior distributions are a remarkably powerful tool available in Bayesian statistics that is not available in Classical statistics. This distribution, which can be constructed using Monte Carlo simulation methods--the ease of which depends on the complexity of the model--provides a thorough view of a parameter of interest. Once constructed, it can be plotted fairly easily in Matlab, R or other mathematical languages. An additional benefit of posterior distributions is their use in solving Bayesian decision theory problems. Jessica Wachter and I used such solutions in our two Journal of Econometrics studies.

Posterior plots are great to show the uncertainty that one has about parameter estimates. I tend to include either a plot of the prior and histogram of the data, or at least point estimates (e.g. the mean) from both, particularly when explaining the analysis. When using conjugate priors, I also include the distribution from which I sampled, for transparency. Also important, although not often published, is a check for convergence. It depends on the sampling method, but I generally plot parameters individually as a time series, using the sample number as time. If the burn-in period was long enough, they should look like a "fuzzy caterpillar" or white noise.

I tend to avoid the 95% "credible interval" because I want to distance myself from the confidence interval. I view my estimated parameters as variables, and the credible interval shows how uncertain I am about them, given my prior and the data. This is different from rejecting a null hypothesis that the true parameter is 0. As a result, I tend to report a smaller range for my credible interval, often 67% or 80%. Since I can have a non-Gaussian posterior for these parameters (e.g. inverse Gamma for variance), they are also not restricted to being symmetric. In the case of variance, this makes sense, as the support for a Gaussian contains negative values, which, for example, a variance cannot take.

In Bayesian statistics and decision analysis, essential visualization tools include Probability Plots (PDFs, CDFs) for distribution insights, Bayesian Posterior Plots (histograms, kernel density) to depict posterior distributions, Trace Plots for MCMC convergence, Decision Trees for decision paths, Sensitivity Analysis Plots (tornado, spider) for parameter impact,(BMA) Plots for model combination, Expected Value Sensitivity Graphs for decision scenario visualization, Credible Interval Plots for uncertainty representation, Bayesian Network Graphs for variable dependencies, and Pareto Fronts for multi-objective decision analysis. These tools enhance the understanding of uncertainties, parameter estimates, and decision outcomes.