import pymc as pm
import numpy as np
import arviz as az

%load_ext lab_black

5. Meta-analysis via Hierarchical Models*

Adapted from Unit 7: rats_nocentering.odc.

Data for x can be found here, and Y here.

y = np.loadtxt("../data/rats_nocenteringy.txt")
x = np.loadtxt("../data/rats_nocenteringx.txt")
x.shape, y.shape

((5,), (30, 5))

This example is taken from Gelfand et al (1990), and concerns 30 young rats whose weights were measured weekly for five weeks.

with pm.Model() as m:
    mu_alpha = pm.Normal("mu_alpha", 0, tau=1e-6)
    tau_alpha = pm.Gamma("tau_alpha", 0.001, 0.001)
    mu_beta = pm.Normal("mu_beta", 0, tau=1e-6)
    tau_beta = pm.Gamma("tau_beta", 0.001, 0.001)

    tau = pm.Gamma("tau", 0.001, 0.001)
    sigma = pm.Deterministic("sigma", 1 / tau**0.5)

    alpha = pm.Normal("alpha", mu_alpha, tau=tau_alpha, shape=(30, 1))
    beta = pm.Normal("beta", mu_beta, tau=tau_beta, shape=(30, 1))

    mu = alpha + beta * x
    pm.Normal("y", mu, tau=tau, observed=y)

    trace = pm.sample(3000)

Show code cell output Hide code cell output

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [mu_alpha, tau_alpha, mu_beta, tau_beta, tau, alpha, beta]

100.00% [16000/16000 00:27<00:00 Sampling 4 chains, 0 divergences]

Sampling 4 chains for 1_000 tune and 3_000 draw iterations (4_000 + 12_000 draws total) took 27 seconds.

az.summary(trace, var_names=["~alpha", "~beta"])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
mu_alpha	106.562	2.294	102.169	110.887	0.020	0.014	13847.0	9366.0	1.0
mu_beta	6.186	0.106	5.982	6.382	0.001	0.001	13038.0	9634.0	1.0
tau_alpha	0.010	0.004	0.004	0.017	0.000	0.000	9855.0	8919.0	1.0
tau_beta	4.313	1.476	1.755	6.912	0.014	0.010	12379.0	9436.0	1.0
tau	0.027	0.004	0.020	0.035	0.000	0.000	7883.0	9493.0	1.0
sigma	6.142	0.467	5.281	7.029	0.005	0.004	7883.0	9493.0	1.0

pm.model_to_graphviz(model=m)

%load_ext watermark
%watermark -n -u -v -iv -p pytensor

Last updated: Wed Mar 22 2023

Python implementation: CPython
Python version       : 3.11.0
IPython version      : 8.9.0

pytensor: 2.10.1

arviz: 0.14.0
numpy: 1.24.2
pymc : 5.1.2