Discrete enumeration in bilinear model

Misc.
1

Hello,
I am a novice with PPL in general and Pyro in particular. I have a very simple model that I am trying to write in Pyro but can’t wrap my head around it.
For context, I have a set of 134 objects, 17 contexts and 17 evaluators.
I have surveyed the evaluators which answered yes/no/unknown to the questions “is object i suitable in context j?” and now I want to model the data and make some inferences.

The model I am considering is pretty straightforward: I am considering a hidden binary z_{ij} variable for each pair (\textrm{object}_i, \textrm{context}_j) which is the true answer to the question “is object i suitable in context j?”, and it is distributed as a \textrm{Bernoulli} variable with parameter p_{ij}. In addition, I have a parameter \alpha_k associated to each evaluator (measuring the expertise or the “adversarialness” of the evaluator) and a parameter \beta_j which models the “difficulty” of each context. I collected each answer in a list y_{ijk} and I am modeling the likelihood of y_{ijk} being 1 as p(y_{ijk}=1|z_{ij}) = \sigma(\alpha_k * \beta_j)^{z_{ij}} \times (1-\sigma(\alpha_k * \beta_j))^{(1-z_{ij})}, where \sigma is the sigmoid function.

This is how I implemented it:

@config_enumerate
def model(object, context, evaluator, y=None):
    n_samples, n_objects, n_contexts, n_evaluators = (
        len(object),
        object.max().item() + 1,
        context.max().item() + 1,
        evaluator.max().item() + 1,
    )
    expertise = pyro.param("expertise", torch.zeros(n_evaluators))
    difficulty = pyro.param("difficulty", torch.ones(n_contexts), constraint=constraints.positive)
    p = pyro.param("p", torch.ones(n_objects, n_contexts) * 0.5, constraint=constraints.unit_interval)
    with pyro.plate("object", size=n_objects, dim=-3):
        with pyro.plate("context", size=n_contexts, dim=-2):
            hidden_label = pyro.sample("hidden_label", dist.Bernoulli(p))
            with pyro.plate("data", size=n_samples) as k:
                e = expertise[evaluator[k]]
                diff = difficulty[context[k]]
                z = Vindex(hidden_label)[object[k], context[k]]
                sigma = torch.sigmoid(e * diff)
                pyro.sample("obs", dist.Bernoulli(probs=(sigma ** z) * ((1-sigma)**(1-z))), obs=y[k])

When I try to run it, I get the error “IndexError: index 100 is out of bounds for dimension 0 with size 2” on the line in which I assign the value to z. I understood that this is likely due to the enumeration taking place along the first dimension.

What would be the correct way of doing this?

2

You should use hidden_label as is without Vindex-ing (z = hidden_label), it has the shape of (2,1,1,1) (https://pyro.ai/examples/enumeration.html). The enumerated values of hidden_label ([0, 1] at dim=-4) will be marginalized out from your model. So if you use TraceEnum_ELBO it will calculate marginalized likelihood: sum(z_ij)[p(z_ij) * p(y_ijk | z_ij)].

3

thanks for taking the time to answer, but if I remove Vindex I still get the same error, and if I remove the indexing from hidden_label altogether I get this shape mismatch error:

Shape mismatch inside plate('data') at site obs dim -3, 16519 vs 2
    Trace Shapes:               
     Param Sites:               
        expertise             17
       difficulty             17
                p         134 17
    Sample Sites:               
      object dist              |
            value         134  |
     context dist              |
            value          17  |
hidden_label dist   134    17  |
            value 2   1     1  |
        data dist              |
            value       16519  |

I should specify that my data y is a long array (16k) of zeros and ones corresponding to the answers, and object, context and evaluator are also arrays of the same length containing the corresponding identities.

4

Judging from the shape of hidden_label being (2,1,1) you are using max_plate_nesting=2. You should set max_plate_nesting=3 in TraceEnum_ELBO since you have three plates (see the tutorial).

5

Then it should only be nested in the data plate.

I think it would help you if you first write down the factorization of your model. Plates correspond to (conditionally independent) products in your factorization, i.e. p(a) \prod_i p(b_i | a) would correspond to

a = pyro.sample("a", ...)
with pyro.plate("i_plate", ...):
    b = pyro.sample("b", ...). # dist depends on 'a'
6

Thank you for the suggestion, indeed it was helpful to go back and give a look at my complete data likelihood. So, the model is supposed to have this structure p(\mathbf{Y}, \mathbf{Z}) = \prod_{i,j}p_{ij}\prod_{k}p(y_{ijk}=1|z_{ij}), where p_{ij} is the probability of the hidden_label z_{ij} being 1. In words, the true latent answers to the question “is object i suitable in context j?” are independent of each other (for now, later I plan to tie all the answers for the same object i with another parameter), then the likelihood of an evaluator answering yes given the latent labels are conditionally independent.
So in order to accommodate for this structure, I rearranged the observations into a tensor of shape y.shape == (134 ,17, 17). Now the model looks like this:

@config_enumerate
def model(y=None):
    n_objects, n_contexts, n_evaluators = y.shape
    expertise = pyro.param("expertise", torch.zeros(n_evaluators))
    difficulty = pyro.param("difficulty", torch.ones(n_contexts), constraint=constraints.positive)
    p = pyro.param("p", torch.ones(n_objects, n_contexts) * 0.5, constraint=constraints.unit_interval)
    mask = ~y.isnan()
    with pyro.plate("object", size=n_objects, dim=-3):
        with pyro.plate("context", size=n_contexts, dim=-2):
            hidden_label = pyro.sample("hidden_label", dist.Bernoulli(p))
            with pyro.plate("evaluator", size=n_evaluators, dim=-1):
                sigma = torch.sigmoid(expertise * difficulty)
                with pyro.poutine.mask(mask=mask):
                    pyro.sample("obs", dist.Bernoulli(probs=(sigma ** hidden_label) * ((1-sigma)**(1-hidden_label))), obs=y)
[...]
svi = SVI(model, guide, optimizer, loss=TraceEnum_ELBO(max_plate_nesting=3))

I used a mask to condition on the observed data because there are missing values in the y tensor, and for now I just want to ignore them and just and let the model see only the observed data, I hope this is the correct way.

However I still get errors, different, but still:

ValueError: Shape mismatch inside plate('context') at site hidden_label dim -2, 17 vs 134
Trace Shapes:       
 Param Sites:       
    expertise     17
   difficulty     17
            p 134 17
Sample Sites:       
   object dist      |
         value 134  |
  context dist      |
         value  17  |
evaluator dist      |
         value  17  |

I don’t get why it messes the dimensions of the hidden_label.

7

That’s probably because the distribution dist.Bernoulli(p) of hidden_label has the batch shape of (n_objects, n_contexts) which correspond to dims (-2, -1) (counting from the right). Pyro plates are used to declare batch dimensions as independent. But they are mis-aligned in your code. For example, pyro.plate("context", size=n_contexts, dim=-2) says that dim=-2 is conditionally independent and has size=n_contexts. But as we saw above hidden_labels distribution has the size of n_objects at dim=-2.

You have to either reshape your distribution or fix plates so that their dims are aligned. There is a great tutorial on shapes in Pyro that explains all of this in more detail.

8

Thank you very much for your explanation! I spent some time (more than I would like to admit!) digging through the code and the tutorials. Starting from simpler models and building up, I managed to implement what I wanted. Even though I feel that this library still hasn’t completely “clicked” yet. But I guess it is a matter of using it more. Anyway, thank you for spending the time in helping me!