Iterative GP prior

Questions v5
1

Hi all,

I am trying to use a GP + softmax as a transition model for hidden states:

with pm.Model() as model:

    ### Prior init state and emission
    p_init_states = pm.Dirichlet("p_init_s", a=[3,2,1])

    # GP state 1:
    l1 = pm.Gamma('l1', alpha=2, beta=1, shape=n_input_s)
    eta1 = pm.HalfCauchy('eta1', beta=5)
    cov1 = eta1 ** 2 * pm.gp.cov.ExpQuad(n_input_s, l1)
    gp1 = pm.gp.Latent(cov_func=cov1)

    # GP state 2:
    l2 = pm.Gamma('l2', alpha=2, beta=1, shape=n_input_s)
    eta2 = pm.HalfCauchy('eta2', beta=5)
    cov2 = eta2 ** 2 * pm.gp.cov.ExpQuad(n_input_s, l2)
    gp2 = pm.gp.Latent(cov_func=cov2)

    # GP state 3:
    l3 = pm.Gamma('l3', alpha=2, beta=1, shape=n_input_s)
    eta3 = pm.HalfCauchy('eta3', beta=5)
    cov3 = eta3 ** 2 * pm.gp.cov.ExpQuad(n_input_s, l3)
    gp3 = pm.gp.Latent(cov_func=cov3)


    ### LL init state and emission
    init_states = pm.Categorical("s_0", p=p_init_states, shape=len(train_ids))

    all_states = pt.set_subtensor(all_states[:, 0], init_states)
    all_states_variables_list.append(init_states)

    for t in range(1, n_timesteps):

        # States transition
        input_state_t = pt.stack([pt.as_tensor(train_ids.ravel()), all_states[:,t-1], pt.as_tensor(action_ts_train.iloc[:, t-1].tolist())], axis=1)
        f1 = gp1.prior(f"f1_{t}", input_state_t)
        f2 = gp2.prior(f"f2_{t}", input_state_t)
        f3 = gp3.prior(f"f3_{t}", input_state_t)

        sm = pm.math.softmax(pt.stack([f1, f2, f3], axis=1), axis=1)
        s = pm.Categorical(f's_{t}', sm)
            
        all_states = pt.set_subtensor(all_states[:, t], s)
        all_states_variables_list.append(s)

        ## ... Emissions process and observations of true emissions...

Now, looking into this topic, I got to understand better how GP latent works and noticed that the way it’s implemented overwrites the given_vals.

Therefore, I am now very uncertain of the quality of this model as I cannot condition on the whole time series observed data.

Did I understand correctly that the current implementation is wrong as it resets the given_vals at every timesteps? Are you then aware of a better implementation using GP?

Many thanks for your help :slight_smile:

2

Hi Christophe!

Constructing loops in PyMC is tricky because we are creating an acyclic graph. You would have to use the scan function to implement what you are trying to do. Have a look at the docs, and see if this helps: scan – Looping in PyTensor — PyTensor dev documentation

3

Thank you for your answer!

I did look into the scan to make it work: it is not trivial to implement a GP inside a scan (scan only accepts tensor as input/output, and so I need to reimplement many functions from GP.Latent inside it).

But anyway, I was more concerned about the GP itself. If I want a single GP to map:

(X_t, some other input) ==> (X_(t+1))

what are the best approaches to do so?

  • If I use iteratively Latent.prior, it will reset all the observed variables of the GP at each timestep (self.X, self.f);
  • would Latent.conditional with hard-coded specific given={“X” : all previous X seen up to timestep t, “f”: all previous output of the GP} be a correct way?
  • And so, would you advise me to implement GP conditional inside the scan function ?

Thanks for your help already,

Chris