Notation

• \(\langle \boldsymbol x, \boldsymbol y \rangle = \sum_{i=1}^N x_iy_i = \boldsymbol x^T\boldsymbol y\)

  • Dot Product; Also covariance of x and y

• \(X \setminus Y\)

  • The set of random variables in set X that are not in set Y

• \(x \bot y\)

  • x is orthogonal to y which means x is uncorrelated with y

• \(x \not\!\perp\!\!\!\perp y\)

  • X is not independent of Y

• \(Y \perp\mkern-10mu\perp X\;|\;Z\)

  • Y is not associated with some variable X, after conditioning on some other variable Z

• \([a, b] \mapsto \mathbb{R}\)

  • Says that \([a,b]\) gets “mapped” to the Reals.

• \(p\;(\;b_i \;∣\; y_t\;;\; \theta \;)\) or \(f\;(\;y \;|\; x \;;\; \theta \;)\)

  • “;” acts as a grand comma to visually emphasize that x is of different kind (e.g. data vs parameters, random vs fixed quantities) than θ
    • i.e. Separates data values from parameters to improve readability

• \(\text{plim}\)

  • Probability Limit
    

    • Interpretations:
      • The value of the random variables gets close to a real number, x, in the sense that the probability that Xn is very different from x gets very small as n gets big.
      • The distribution of Xn gets very close to the distribution of some other random variable Y as n gets large (then we would need a definition for the distance between distributions).