Question on MAP posterior approximator

[PaulScemama]

Hi! I trained a MAP posterior approximator and was looking at predictive samples:

map_model.prective.sample(data_loader.to_inputs_loader())

And I observed that each time I call this I get different answers. Where is the randomness coming from if a MAP approximation only sees the parameters $\mathbf{\theta}$ point-estimates? I looked at the documentation but couldn't find anything.

Thanks in advance!

[gianlucadetommaso]

Hi Paul, you are sampling target variables $y^{(i)}$ from the predictive distribution $p(y|x, D)$, where $x$ is a certain input for which you want to get target samples, and $D$ is your training data. In order to sample from the predictive distribution, you can actually sample from the joint distribution $p(y, \theta|x, D) = p(y|\theta, x)p(\theta|D)$, where $\theta$ are model parameters. This means that, in order to obtain a sample $y^{(i)}$ given $x$ and $D$, you can first sample $\theta^{(j)}$ from the posterior $p(\theta|D)$, then sample $y^{i}$ from the likelihood $p(y|\theta^{(j)}, x)$.

As you correctly said, there is no randomness in sampling from the MAP approximation of the posterior, since you always get the sample parameters $\theta^{MAP}$. However, there is still randomness when sampling from the likelihood $p(y|\theta^{MAP}, x)$.

I hope this explains it. Please let me know if it is clear.

[PaulScemama]

Thank you very much for the clear explanation! This makes sense, thank you :). One follow-up question: is the randomness when sampling from the likelihood from a fixed variance that's calculated (i.e. $\beta$?) In other words, is it assuming fixed variance due to (e.g.) label noise?

[gianlucadetommaso]

From your question, I assume you are doing regression? In such case, the likelihood variance model is also learned (unless you don't want to, in which case you can just pass a hyperparameter). In ProbRegressor you need to pass both model, which is the mean likelihood model, and likelihood_log_variance_model, which is the log-variance of the likelihood. If the latter is a constant model, then you would just get a constant.

[PaulScemama]

I'm doing classification, but the question was just out of curiosity. Very cool! Thanks for such clarification. Feel free to close