Reversing "mean-centered" parameters in a multiple linear regression

Question

I tried searching for this question on stats stack exchange and found Implementing linear regression with standardization but the answer was a little difficult to follow. I'm reading "Bayesian Analysis with Python" by Osvaldo Martin (great read btw) and in his hierarchical linear models section he often mean-centers the data and the reverses it. Can somebody please explain this process to me and how to rearrange the values to visualize the reversal after mean-centering? The line that is confusing me is alpha = pm.Deterministic("alpha", alpha_tmp - pm.math.dot(beta, X_mean)) why does subtracting the dot product of the betas and the mean from the alphas reverse the mean centering? I feel like I'm missing something very simple.

The author implements it in Python 3.5 using a module that is up and coming called pymc3. Here is the code excerpt below:

alpha_tmp is the alpha when X is mean centered. The formula that is being used is:

$$\mu = \alpha + \beta_1*x_1 + \beta_2*x_2$$

import pymc3 as pm
import numpy as np

# Multiple Linear Regression
# pg. 132
np.random.seed(314)
N = 100
alpha_real = 2.5
beta_real = [0.9, 1.5]
eps_real = np.random.normal(loc=0, scale=0.5, size=N)

X = np.array([np.random.normal(i,j, N) for i,j in zip([10,2],[1,1.5])])

X_mean = X.mean(axis=1, keepdims=True)
X_centered = X - X_mean
y = alpha_real + np.dot(beta_real, X) + eps_real

with pm.Model() as model_mlr:
    alpha_tmp = pm.Normal("alpha_tmp", mu=0, sd=10)
    beta = pm.Normal("beta", mu=0, sd=1, shape=2)
    epsilon = pm.HalfCauchy("epsilon", 5)

    mu = alpha_tmp + pm.math.dot(beta, X_centered)

    alpha = pm.Deterministic("alpha", alpha_tmp - pm.math.dot(beta, X_mean))

    y_pred = pm.Normal("y_pred", mu=mu, sd=epsilon, observed=y)

    start = pm.find_MAP()
    step = pm.NUTS(scaling-start)
    trace_mlr = pm.sample(5000, step=step, start=start)

varnames = ["alpha", "beta", "epsilon"]
pm.traceplot(trace_mlr, varnames)

# Below is output of stderr
Optimization terminated successfully.
         Current function value: 74.986175
         Iterations: 23
         Function evaluations: 31
         Gradient evaluations: 31
100%|██████████| 5000/5000 [00:13<00:00, 380.52it/s]

Answer 1

The answer by John Kruschke is the right one. And is also explained on page 102 of "Bayesian Analysis with Python" (BAP). Since the explanation in BAP is very brief, and probably not clear enough (sorry about that!) I will give here a little bit more detail:

We get $x'$, the centered version of $x$ by doing:

$$x' = x - \bar x$$

Then we can write the linear model as:

$$y = \alpha' + \beta' x' + \epsilon$$

if we replace $x'$ with $x - \bar x$, we get:

$$y = \alpha' + \beta' (x - \bar x) + \epsilon$$

reordering we get:

$$y = \alpha' - \beta' \bar x + \beta' x + \epsilon$$

Notice that this last equation is equivalent to

$$y = \alpha + \beta x + \epsilon$$

where:

$$\alpha = \alpha' - \beta' x$$

and

$$\beta = \beta' $$

Answer 2

It's explained on p. 485 of Doing Bayesian Data Analysis Second Edition.

Just start with the predictiin equation for mean centered parameters, substitute the mean centered variables, and reexpress in terms of the original variables.