Bayesian Hierarchical Models

By Žan Pušenjak

Bayesian Statistics, FRI, University of Ljubljana

Dec 28, 2025
bayesian-statistics

Introduction

We were tasked to answer question about global happiness levels based on two provided datasets. The cantril ladder ¹ and country GDP and life expectancy ².

Combined the datasets gave us information about country level happiness scores over the past few years. The data consisted of five features country, continent, year, log_gdp (logarithm of the GDP, original unit (before log) was US dollars) and life_expectancy and a target variable score that ranged from 0 (maximum sorrow) to 10 (maximum bliss), meaning that 5 is the neutral boundary between happiness and unhappiness.

Quickly inspecting the data we observed that it was very lacking (for the amount of parameters we were tasked to infer from it) since some countries only had one or two years of data effectively giving us one datapoint for that country.

Methods

Data preparation

After joining the two datasets we first encoded the continents and countries with indexes ranging from 1-6 for continents and 1-165 for countries for easier usage.

Next we standardized the year, log_gdp and
life_expectancy with $(x - \mu_x){\sigma_x}^{-1}$ .

Modeling

We used a hierarchical bayesian linear model with two hierarchical links. Here we must note that the link were horizontally stacked, so the model only had one level of hierarchy, which simplified the model as opposed to building a full tree level hierarchical model, giving us a better fit with the given data.
Our model ended up being comprised of three parts:

Global level link - we used the global hierarchical link for year coefficient and intercept
$\beta_{year\_i} \sim \text{N}\!\left(\mu_{year},\, \sigma_{year}\right)$ and $\alpha_i \sim \text{N}\!\left(\mu_{alpha},\, \sigma_{alpha}\right)$
Continental level link - for all other coefficients we used a continent level hierarchical link
$\beta_{gdp\_i} \sim \text{N}\!\left(\mu_{gdp\_j},\, \sigma_{gdp\_j}\right)$ and $\beta_{life\_i} \sim \text{N}\!\left(\mu_{life\_j},\, \sigma_{life\_j}\right)$

Where $i$ is the index of a country and $j$ is the index of the continent that the country $i$ is located in.
Country level prediction $y \sim \text{N}\!\left(a_i + X\beta,\, \sigma\right)$ where $\beta$ is a vector $(\beta_{year\_i}, \beta_{gdp\_i}, \beta_{life\_i} )$

Where $i$ is again the index of the country.

Because of the large amount of parameters and lacking data we had to alter some of the sampling parameters to get a good fit.
We ran 4 chains for 5000 iterations with a 2000 iteration warmup. We increased the adapt delta to $0.95$ and set the maximum tree depth to $15$ .

Diagnostics

After training we checked some of the trace plots and $\hat R$ , both of which looked fine so the training was successful.

Results

Global analysis

Looking at the global $\mu_{year}$ (Figure 1) we can be $89.250\% \pm0.297\%$ certain that happiness scores are increasing yearly. Which is a good sign for us.

Posterior distribution of the $\mu_{year}$ with 90% CI

Furthermore plotting the posterior predictive of the global happiness in 2025 (calculated $y = \alpha +x_{year}(2025)\beta_{year\_global}$ where $x_{year}(2025)$ is the standardized value of 2025) we can see that we can be as sure as possible that people globally are happy, meaning the happiness score is above $5$ (Figure 2). Again a pleasant finding.

Prediction of global happiness in 2025 with 90% CI

Slovenia ranking

We wanted to know if Slovenia is currently (meaning in 2025) in the top 20 of the happiest countries in the world. The probability of this being true is $39.916\% \pm 0.424\%$ .

Plotting the top 20 happiest countries (based on their mean happiness score value) we can further observe, how Slovenia compares to them (Figure 3)

Slovenia compared to top 20 happiest countries

Continental differences

Lastly we looked at how life expectancy and GDP are correlated with the happiness scores across different continents. More specifically we searched for any outstanding differences across the continents.
Firstly observing Figure 4 we can see that Africa and Europe distribution means are roughly the same but Africa has a much smaller variance. The happiness in these two continents seems to be loosely positively correlated with life expectancy.
North America and Asia have the most extreme cases of correlation, where Asia is negatively correlated with happiness and North America being the most positively correlated.
Lastly Oceania and South Africa, similarly as Africa and Europe, have similar distribution means but what stands out with these two continents is their variance, Oceania in particular has very large variance.

Correlation of life expectancy across continents

With GDP plot (Figure 5) the picture stays mostly the same.
Africa still has small variance and a mean that shows slight positive correlation.
The biggest difference is Europe that almost matches Asia distribution, which is again one of the two extreme cases of correlation. Interestingly enough the positive and negative correlation with the happiness score now swaps so North America is the most negatively correlated and Asia (and also Europe) being the most positively correlated.
Like with life expectancy Oceania has a very wide distribution with a large variance and a similar increased variance can be observed in South Africa.

After analysing the results we can see that there are significant differences between the correlation of health and wealth, not only that, the correlation can be very different among those two variables. Africa is the most consistent with their happiness because of the small variance. With this continent we also see that health and wealth do not play a very significant role on the overall happiness.
In Europe it seem that health likewise does not contribute a lot to the happiness scores, but the wealth seems to be very important to europeans.
North America and Asia seem to be mirroring one another. In Asia a longer life expectancy lowers the happiness scores. This could hint that the older people are more miserable and younger asians are happier. On the flip side, like with Europe, wealth is very positively correlated with happiness. With North America it is just the opposite. That is interesting especially in the case of wealth which is negatively correlated with happiness.
South America is similar to Africa in regards to the wealth. While the variation is higher, the correlation is low and even slightly negatively correlated. Health in South America is an important factor since it is positively correlated with happiness, but not to the extend as in North America.
Lastly Oceania has a very large variation in both health and wealth correlations which hints very large differences in the life and happiness reasons of its people.

Discussion

Using a one layer hierarchical model with two different links gave us a better fit with the provided data. The predispositions of a global yearly trend and global intercept are quite strong, but given the questions we were trying to answer and the lacking datapoints of some countries it would be unnecessary to try and fit a full two layer hierarchical model.

The positive yearly trend and a larger than $5$ global happiness score was a very welcome discovery.

https://github.com/fri-datascience/course_bs/blob/main/
session_07_hierarchical_models/data/cantril_ladder.csv↩︎
https://github.com/fri-datascience/course_bs/blob/main/
session_07_hierarchical_models/data/countries.csv↩︎