Bernoulli correlations

Questions
1

Hi, All
For example, | have a two biased coins where Bernoulli parameters of their is unknown for me.
But I know that both of it depend on the single latent variable.
In general, I have P(θ1|_latent) and P(θ2|_latent) distribution, but, for simplification,
lets dependency will be linear, so θ1 = link(_latentc1), θ2 =link(_latent c2), where c1 and c2 constants
is known.
Now, I got 100 trials for first coin, and 50 for second. How I can join that information to infer the _latent parameter ?

2

That’s fine, you will have two likelihood, two expressions for the thetas, but one distribution/model for the hierarchical latent variable. Since the latent variable is at the top of the graph for both likelihoods, it will be influenced by both.

3

Thank you! But, it can somehow be described as pymc3 snippet?

4

I think something like this is what you want:

import pymc3 as pm
import numpy as np
import pylab as pl
from scipy.special import expit

## Generate data
N0, N1 = 1000, 500
true_latent = 0.2
c0, c1 = 1., 5.
true_p0 = expit(c0 * true_latent)
true_p1 = expit(c1 * true_latent)

data = [(np.random.rand(N) < p).astype(np.int) for p, N in zip([true_p0, true_p1], [N0, N1])]

## Define model
with pm.Model() as model:
    latent = pm.Normal('latent', 0., 1.)
    c0 = pm.Exponential('c0', 1.)
    c1 = pm.Exponential('c1', 1.)
    p0 = pm.Deterministic('p0', pm.math.sigmoid(c0 * latent))
    p1 = pm.Deterministic('p1', pm.math.sigmoid(c1 * latent))
    pm.Bernoulli('x0', p0, observed=data[0])
    pm.Bernoulli('x1', p1, observed=data[1])

## Inference
with model:
    trace = pm.sample(1000, target_accept=0.99)

Note that this particular code doesn’t yield a good fit because the model allows ambiguities between latent variable and the constants c0, c1. But in principle, I believe that this does what you’re asking for:

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