Learning From Data
Business Analytics
Y
X
All U.S. Houses
Price
Size
\mathbb{E}[Y \vert X]
Sample Size
Features
Neighbors
Y
X
Sample Space
SET A
SET B
(Countable)
(Uncountable)
Y
\mathbb{E}[Y \vert X]
All U.S. Houses
Price
Price
Price
\varepsilon
=
+
Y_i = \mathbb{E}[Y \vert X=X_i] + \varepsilon_i
Price of House
Average Price Given Its Size
Error Term
Y_i = \beta_0 + \beta_1\text{Size}_i + \eta_i
Price of House
Slope
Error Term
Y-intercept
\{(Y_i, X_i)\}_{i=1}^n
Observed
f(\theta, X)
Build & Fit
\mathbb{E}[Y\vert X]
Interest
Setup
Learning From Data
(1) Data
(2) Function Space
Parameter Space
(3) Objective Function
(4) Solver
(Build & Fit)
\varepsilon_i
\eta_i
\hat{\eta}_i
Y
X
All U.S. Houses
Price
Size
\mathbb{E}[Y \vert X]
\frac{\frac{1}{n}\sum_{i=1}^n (\hat{Y}_i - \bar{\hat{Y}})^2}{\frac{1}{n}\sum_{i=1}^n (Y_i - \bar{Y})^2}
Data
Model
1
\text{Var}(\bar{X}) = \frac{\text{Var}(X)}{n}
2
\frac{1}{n-1}\sum_{i=1}^n(X_i - \bar{X})^2 \approx \text{Var}(X)
3
\frac{1}{n(n-1)}\sum_{i=1}^n(X_i - \bar{X})^2 \approx \frac{\text{Var}(X)}{n}
f_x
Set of Parameters
\big( \mathcal{R}^d, \mathcal{B}(\mathcal{R}^d), \mathbb{P}_n \circ \mathcal{A}_n^{-1}\big)
\big( \mathcal{R}, \mathcal{B}(\mathcal{R}), \mathbb{P}_n \circ \mathcal{A}_n^{-1} \circ f_x^{-1}\big)
Set of Outcomes
\mathcal{A}_n
Set of Possible Data Sets
\big( \Omega_n, \mathcal{F}_n, \mathbb{P}_n\big)
\text{MSE}
Set of Parameters
\big( \mathcal{R}^d, \mathcal{B}(\mathcal{R}^d), \mathbb{P}_n \circ \mathcal{A}_n^{-1}\big)
\big( \mathcal{R}, \mathcal{B}(\mathcal{R}), \mathbb{P}_n \circ \mathcal{A}_n^{-1} \circ \text{MSE}^{-1}\big)
Set of Outcomes
\mathcal{A}_n
Set of Possible Data Sets
\big( \Omega_n, \mathcal{F}_n, \mathbb{P}_n\big)