Prior on parameters different than the proposal prior (SNPE_C with existing simulations)

[CompiledAtBirth]

Hello,

I am experimenting with the sbi package in the specific case of SNPE_C -- Greenberg, Nonnenmacher & Macke (ICML 2019) -- with summary statistics from existing simulations. The feedback of advanced users of simulation based inference can be very interesting.

Since the forward model is intensive and I cannot sample summary statistics from it on the fly (here N-body cosmological simulations) , I am using a single-round of Algorithm 1 on the whole dataset.

The parameters of the pre-run simulations have been sampled from a Latin-Hypercube, corresponding to a specific BoxUniform prior which is by definition my proposal prior.

As far as I know, we can also pass the proposal prior through the proposal argument in the append_simulations method defined in snpe_base.py.

proposal: The distribution that the parameters $\theta$ were sampled from.
            Pass `None` if the parameters were sampled from the prior. If not
            `None`, it will trigger a different loss-function.

What could be good practice when using $p_1^{\sim}(\theta) \neq p(\theta)$, where $p_1^{\sim}$ is the proposal prior of the simulations dataset and $p(\theta)$ a specific prior modeling previous knowledge about the parameters ?

Below are some plots from a toy test on 3 parameters with a standard summary statistic in Cosmology.

• The "prior=proposal" case is the "correct" (green) case I would assume I am limited to, since I cannot sample new simulations (the prior is the same BoxUniform prior as from the existing statistics were sampled from)
• "MVN prior, proposal ignored" means I am using a MultiVariate Normal prior for the parameters, but I do not pass the proposal prior argument to the SNPE_C instance. This case should be "wrong" since my simulations were not sampled from this distribution and Algorithm 1 considers the MVN prior as the proposal prior.
• "MVN prior + correct proposal" (in red) means I instantiate SNPE with prior= $p(\theta)$ but pass a different proposal prior $p_1^{\sim}(\theta) != p(\theta)$ in the append_simulations method.

As illustrated in the plots, using an inaccurate proposal "MVN prior, proposal ignored" does not seem to impede the training and constraints too much, but using a prior different than the accurate proposal distribution clearly shows something is going wrong (eg corner plot in red).

Has the $p_1^{\sim} \neq p$ case been documented and tested in the SNPE literature?

NB: I am using a simple Mixture Density Network (MDN) as the density estimator defined through utils.get_nn_models.posterior_nn