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]

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]

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:

$$\mu = \alpha + \beta*x_1 + \beta*x_2$$

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]

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]