Monte Carlo estimate / scale factor for likelihood

1

Assume that I have a model like: f ~ Normal(a, b), y ~ Bernoulli(f). How to write a model in Pyro to get likelihood estimate E_{p(f)} log p(y|f) using MC with 10 samples of f? For 1 sample we can write like this

def model(y):
    f = Normal(a,b)()
    return pyro.sample("y", Bernoulli(f), obs=y)

Now, I build a guide q(f) for f and want to use an objective like KL(q(f)||p(f)) + alpha E_{q(f)} log p(y|f) where alpha is a scale factor. Is there a way to add a scale factor without rewrite ELBO? This is useful for training models with mini-batch.

2

For those who interested in this, we can obtain these targets by using pyro.poutine.scale decorator. :slight_smile:

2 Likes
4

Sorry to dig up this old post. But does pyro.poutine.scale solves the first problem in the original post? If I understand correctly f here is not a latent variable and the likelihood is the E_{p(f)} log p(y|f) which is itself an intractable intergral and can be esimated as you suggested by a MC estimator. Can pyro.poutine.scale actually do this, or maybe we need pyro.poutine.mc_extend to do the job?

5

Side question: Shouldn’t f be constrained to be a fractional? It wouldn’t be a valid Bernoulli parameter otherwise, right?

6

We can draw a batch of f (say 10) and then scale likelihood to 1/10.

7

Yes, you are right. A response function (inverse link function) should be used here

8

Thanks. So if I understand correctly, for the unscaled model, when writing pyro.sample statement for observables, we need to expand each data point with the number of samples used to estimate the likelihood (10 in this case). Correspondingly we need to sample 10 samples of non-latent variable f in the model using torch.distributions instead of pyro.sample. Is this the correct way to do it? Or is there a more poutine way of doing it? Thanks.

9

I’ll do that way because I have some specific reason. I don’t know if there is a poutine way to do it.

I guess you can set num_particles=10 in ELBO to get what you want.

10

Thanks. Setting num_particles=10 is probably not what I want since if I understand correctly this means how many do you sample from the guide. In my case (or this case) the likelihood itself is an integral and needs multiple samples to estimate. At least using torch.distribution is one correct way to do it.