Conditional multivariate sampling

Questions
1

Hello,

I fitted a multivariate normal distribution over a rhee dimension matrix of random variables using the following code:

model = pm.Model()
shape = x.shape[1] # shape of x is [N, 3]

with model:
    packed_L = pm.LKJCholeskyCov('packed_L', n=shape,
                                 eta=2., sd_dist=pm.HalfCauchy.dist(2.5))
    L = pm.expand_packed_triangular(shape, packed_L)
    Σ = pm.Deterministic('Σ', L.dot(L.T))
    
    μ = pm.Normal('μ', 0., 10., shape=shape)
    
    obs = pm.MvNormal('obs', mu=μ, chol=L, observed=x)
    trace_ = pm.sample(5000, cores=1)

I would like to fix one variable of the distribution to a scalar value and have the resulting conditional distribution of the two remaining variables. How can that be achieved?

Thank you very much for your help.

Maxime.

2 Likes
2

You mean like fixing μ[0] = scalar with the observed being the same observed=x? or you would like to marginalized the MvNormal to have the observed=x[:, 1:]?

3

Thank you for your answer.

Let’s say that X1 is a random variable describing a person’s age, X2 is the annual revenue and X3 represents this person’s number of children.

The objective of the first code snippet was to model these variables with a multivariate normal distribution. Now, I would like to have the normal multivariate distribution of say X1 and X2 given that X3=3 children. Would that be possible ?

x = [X1, X2, X3]

Thank you in advance,

Maxime.

1 Like
4

Since [X1, X2, X3] are some observed quantities, I would either model x_tilt = [X1[X3==3], X2[X3==3]] and find the posterior of x_tilt; or model X = [X1, X2, X3] directly and find the posterior of X, then compute the conditional of this posterior when X3==3.

The later cases you can use the code you have, and use sample_ppc after inference to generate large among of ppc samples, and plot the conditional:

index = (ppc['obs'][:, 2] > 2.99)  or (ppc['obs'][:, 2] < 3.01)
scatter(ppc['obs'][index, 0], ppc['obs'][index, 1])
1 Like
5

As X3 represents a continuous variable in my model, I will use the second method.

Thank you for your help.

Maxime.

6

FYI, using the second approach you can also compute the posterior conditional using the posterior sample mu and cov of the MvNormal from the trace: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions

1 Like