Posterior estimation & inference
1. Posterior estimation & inference
Our work thus far has focused on the definition of Bayesian models and the RJAGS simulation of their corresponding posteriors. A lot of hard work remains. In this video we'll explore the construction of numerical summaries that communicate our simulation results, provide estimates of posterior trends, and can be used to make inferences about the broader population of interest.2. Bayesian regression model
First, recall our Bayesian regression model of weight $Y_i$ versus height $X_i$. In light of our prior models of the regression parameters $a$, $b$, and $s$ as well as the observed data on 507 adults, we simulated a posterior regression model using RJAGS.3. Posterior point estimation
For simplicity, we'll focus on the 100,000 iteration simulation results for the intercept $a$ and slope $b$ parameters. The approximate posterior density plots constructed from the simulations are shown here. In a Bayesian analysis, we can think of the *entire* posterior densities as estimates of $a$ and $b$. After all, these collections of posterior plausible values provide a complete picture of the posterior trends and uncertainty in our parameters. However, in understanding and communicating our posterior model, it can also be useful to provide simple summaries of the posterior models.4. Posterior point estimation
For example, the posterior means of $a$ and $b$, marked by the red vertical lines, provide simple point estimates of the posterior trends in these parameters.5. Posterior point estimation
A `summary()` of our mcmc.list object `weight_sim_big` returns the estimates of the posterior means. These are simply the means of the Markov chain values and can be found in the `Mean` column of the first table. Here, the posterior mean of $y$-intercept $a$ is approximately -104.038 kg and the posterior mean of slope $b$ is approximately 1.012 kg/cm.6. Posterior point estimation
Combined, the posterior means of the intercept and slope provide an estimate of the posterior mean trend in the relationship between weight and height. However, keep in mind that this is simply the *mean* trend. We observed 100,000 plausible sets of intercept $a$ and slope $b$ values in our `weight_chains`.7. Posterior uncertainty
Plotting the lines that correspond to just the first 15 sets gives us a sense of the overall posterior *uncertainty* in the trend. It's important to reflect this uncertainty in our numerical summaries of the posterior.8. Posterior credible intervals
A common approach is to calculate posterior credible intervals. The ranges of the red regions here correspond to 95% credible intervals for $a$ and $b$. They reflect the middle 95% of posterior plausible parameter values, thus communicate our uncertainty in the parameters.9. Posterior credible intervals
Estimates of the 95% credible intervals for each model parameter are reported in Table 2 of the `weight_sim_big` summary. The lower and upper limits of the intervals are simply calculated by the 2.5th and 97.5 quantiles of the corresponding Markov chain values. Here, the 95% posterior credible interval for $a$ ranges from -118.6843 to -87.470 kg. The 95% posterior credible interval for $b$ ranges from 0.9152 to 1.098 kg/cm.10. Posterior credible intervals
The interpretation of credible intervals is also refreshingly intuitive. Consider the interval for slope $b$. We can say that, in light of our priors & observed data, there's a 95% posterior chance that $b$ is between 0.9152 & 1.098 kg/cm.11. Posterior probabilities
Finally, we can calculate numerical summaries that inform specific hypotheses. For example, suppose we're interested in whether the increase in weight per 1 cm increase in height exceeds 1.1 kg. The posterior probability of this event is represented by the area of red region here. An examination of our simulation data reveals that only 2,165 (or 2.165%) of the 100,000 $b$ Markov chain values exceed 1.1. Thus, we estimate that there's only a 2.165% posterior chance that slope $b$ exceeds 1.1 kg/cm.12. Let's practice!
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