Using Posterior Samples From First in Second Model

numpyro
1

Hi everyone!

I’m trying to use the posterior samples of a model during the sampling of model2; Currently I simply “glue” both models together, using 2 sampling statements with different obs= but there should be a better way, right?

A simple example would be the following:

import numpyro
import numpyro.distributions as dist
import numpy as np


# Gen Data
x = np.random.normal(0, 1, 100)
m = 5
b = 2
y = m * x + b + np.random.normal(0, 1, 100)

scale = 2.
y2 = y * scale


def model(x):
  m = numpyro.sample("m", dist.Normal(0,1))
  b = numpyro.sample("b", dist.Normal(0,1))
  
  with numpyro.plate("data", len(x)):
    numpyro.sample("y", dist.Normal(m * x + b, 1), obs=y)

kernel = NUTS(model)
mcmc1 = MCMC(kernel, num_warmup=1000, num_samples=1000, num_chains=4, progress_bar=True, chain_method='parallel')
mcmc1.run(rng_key_,x=x)
mcmc1.print_summary()

def model2(x):
    m = numpyro.sample("m", dist.Normal(0,1))
    b = numpyro.sample("b", dist.Normal(0,1))

    projection = (m * x + b)
    numpyro.sample("y", dist.Normal(projection, 1), obs=y)

    scale = numpyro.sample("scale", dist.Normal(0,1))
    numpyro.sample("y_maxed", dist.Normal(projection * scale, 1), obs=y2)

kernel2 = NUTS(model2)
mcmc2 = MCMC(kernel2, num_warmup=1000, num_samples=1000, num_chains=4, progress_bar=True, chain_method='parallel')
mcmc2.run(rng_key_, x=x)
mcmc2.print_summary()

While model2 works (in the sense that the mean parameter values of m and b are correct, I noticed that the effective number of samples in model_2 for m and b is lower (which makes sense) and this kind of double-sampling seems error-prone.

Happy to get any hints!

Best
N

2

what is the generative process? why are you effectively observing the same thing twice?

3

Thanks for the reply!

The idea is that I observe the data on y2 and I have an assumption on how this y2 comes into being, namely as a function of some intermediate value y (here very simply just y2 = f(y) = 2*y) for which is also observe data; in other terms, modeling y could be called a “measurement model”, whereas I’m ultimately interested in the data of y2.

4

i’m afraid i don’t understand. are both y and y2 observed/measured in the sense that you have concrete values for each of them?

5

Yes exactly, the idea is that I can measure parts of the whole system and the lets say final outcome and want to model everything jointly.

6

if you have two measurements that are independent in some sense—one taken with a microscope, one with a scale, say—then having two observe statements would generally be appropriate