# what is the correct way to compare log likelihood of various models?

I am trying to compare numerous neural network models in terms of log likelihood, for example there are two given here...

In the first example, they use the logsumexp trick in order to calculate the log probability over the number of samples $$K$$ they take (https://github.com/yaringal/DropoutUncertaintyExps)...

$$\log p(y_i | x_i) = \log \frac{1}{K} \sum_k \exp{\log{p(y_{ik} | x_{ik})}}$$

Which measures the average log probability over each sample. This gives a final log probability but we do not have the parameters a density to go along with this distribution (as far as I know). So if I wanted to graph out the mean and standard deviation of this, I could not do it.

On the other hand, the second paper I posted calculates log probability by taking a mixture of Gaussians for the predictions from different models (similar to sample $$K$$ above) and then calculates the log probability based on this resulting density.

$$\mu_* = \frac{1}{K} \sum_k \mu_k \\ \sigma^{2}_* = \frac{1}{K} \sum_k(\sigma^2_k + \mu^2_k) - \mu_*^2 \\ \log p(y_i | x_i) = \log \mathcal{N}(y_i| \mu_*, \sigma^2_*)$$

Which gives a different numerical result than the first method.

In my opinion, I think the second method is better because in a real test situation, we need to have a final density in order to evaluate the prediction of the model, right?