I will assume readers’ awareness of the basics to an extent that readers know what keywords to search for. I will lay out necessary mathematics / computer science knowledge in this page, such that readers can focus on how to apply these math/CS knowledge to think about cognition when reading the note:

Notes on the Bayesian Models of Cognition

Basic Awareness

• CHP 395: Thinking and Reasoning
• Naïve Bayes Classifier
• Statistics for Psych/AI

Notation

• $A \subset B$ means solely that A is a subset of B. A may or may not be equal to B. The notation $\subseteq$ is avoided.
• $[.]$ is the Iverson bracket.
• Def. Gamma function $\Gamma(r)=\int_0^\infty y^{r-1} \exp(-y) dy$.
  • When r is an integer, $\Gamma(r) = (r-1)!$
  • $\Gamma(r) = (r-1) \Gamma(r-1)$
• Def. Beta function $B(r,s)=\int_0^1 \theta^{r-1} (1-\theta)^{s-1} d\theta = \frac{ \Gamma(r) \Gamma(s) }{ \Gamma(r+s) }$
• Def. Sigmoid/Logistic function $sigmoid(x) = \frac{1}{1 + e^{-x}}$

Math: Probability

Bayesian Inference

• When there are only two hypotheses, work with posterior odds is easier.
  • Take the ratio between two formulas of the Bayes’ rule
  • $\frac{Pr(h_1|d)}{Pr(h_0|d)} = \frac{Pr(d|h_1)}{Pr(d|h_0)} \frac{Pr(h_1)}{Pr(h_0)}$
  • Log likelihood ratio $\log \frac{Pr(d|h_1)}{Pr(d|h_0)}$ is often taken as a measure of the evidence that d provides for $h_1$.
  • A convenient consequence is that $Pr(h_1|d)=\text{sigmoid}(\log \frac{Pr(d|h_1)}{Pr(d|h_0)} + \log \frac{Pr(h_1)}{Pr(h_0)})$
• Discrete hypotheses and observations
  • Likelihood matrix L. A column represents a hypothesis. A row represents a possible observation, such that $L_{ji}=P(d=j|h_i)$.
  • Implement Bayesian updating as adding a likelihood row vector (from L) to a prior row vector.
  • (softmax) Remember to subtract the maximum value from entires representing the log probabilities when converting back to probabilities.
• Continuous hypothesis space

  • Case study: estimating the bias $\theta$ of a coin (p. 78)

    • The data $d$ is a sequence of $n_H$ heads and $n_{T}$ tails.
    • Frequentist approach: maximum-likelihood estimation
      • $Pr(d|\theta) = \theta^{n_H} (1-\theta)^{n_T}, \hat \theta = \argmax_\theta Pr(d|\theta) = {n_H \over n}$
    • Bayesian approach: obtain posterior distribution $Pr(\theta|d)$ + [posterior mean/maximum a posteriori]
      • Likelihood = the previous Bernoulli $Pr(d|\theta) = \theta^{n_H} (1-\theta)^{n_T}$
      • Prior = let’s say uniform for now.
      • $Pr(\theta|d) = { Pr(\theta)Pr(d|\theta) \over \int Pr(\theta')Pr(d|\theta') d\theta' } = { \theta^{n_H} (1-\theta)^{n_T} \over B(n_H+1,n_T+1) } \sim Beta(n_H+1, n_T+1)$
      • Def. Posterior mean $\bar \theta = \int \theta\ Pr(\theta|d) d\theta = \frac{n_H+1}{n_H+n_T+2}$. A weighted average.
      • Def. Maximum a posteriori (MAP). $\hat \theta = \argmax_\theta Pr(\theta|d)={n_H \over n}$

  • Def. Conjugate priors

    • Beta distribution is the conjugate prior for the Bernoulii or binomial likelihood.
    • E.g., in the coin flipping game, if $Pr(\theta)\sim Beta(v_H+1,v_T+1)$ with $v_H, v_T>0$, then $Pr(\theta|d) \sim Beta(v_H+1+n_H, v_T+1+n_T)$, as if $v_H, v_T$ represent the number of virtual examples and both the real and virtual examples had been observed in the same dataset. This can be extended to $Pr(\theta)\sim Beta(\alpha, \beta)$ with $\alpha, \beta \in (0, \infty)$.

    Prior	Likelihood	Posterior
    Beta	Bernoulli/Binomial	Beta
    Normal	Normal	Normal
    Gamma	Poisson	Gamma
    Inverse Gamma	Normal Variance Estimation	Inverse Gamma

Special Distributions

• Single draw

  • Def. (Bernoulli distribution, $X \sim Be(p)$ ) Bernoulli trial is that $A \in \mathcal{F}$, and we call the trial a success if $A$ occurs. Bernoulli distribution is based on single Bernoulli trial.
  • Def. (Categorical distribution/multinoulli distribution) The possible result can take on one of $K$ possible categories, unlike 2 categories in Bernoulli distribution. “The categorical distribution is a special case of the multinomial distribution, in that it gives the probabilities of potential outcomes of a single drawing rather than multiple drawings.” [1]
  • Def. (Beta distribution $X\sim Beta(\alpha,\beta)$ )

    • $f_X(x) = { x^{\alpha-1} (1-x)^{\beta-1} \over B(\alpha,\beta) }$

    • Prefer the (mean, precision) formulation.

      

• Multiple draws

  • Def. (Binomial distribution, $Y \sim Bin(n,p)$ )
    • Perform n independent Bernoulli trials with $p=\text{I\kern-0.15em P}(A)$, and let Bernoulli r.v.s. $X_{1}, X_{2} \dots X_{n}$ be the indicator function of success of the experiments.
    • Let $Y:=\sum X_{i}$, then the p.m.f. $f_{Y}(k)={n \choose k} p^{k} (1-p)^{n-k},k=1, 2, \dots, n$.
    • Def. Binomial process. $Y_n$ is the number of successes in the first n $Be(p)$ trials.
    • Def. (Geometric distribution, $W \sim Geom(p)$). Keep performing Bernoulli trials until the first success. Let $W$ denotes the waiting time. $f_W(k) = (1-p)^{k-1}p, F(x)=1- (1-p)^x$.
    • Def. (Negative Binomial distribution, $W_r \sim NB(r,p)$). $W_r$ is the waiting time for $r$ successes.
  • Def. (Beta-Binomial distribution, $Y \sim BetaBin(n, \alpha, \beta)$)

    • Three equivalant formulations:

      1. Draw $p \sim Beta(\alpha, \beta)$, then $Y \sim Bin(n,p)$
      2. For i from 1 to n, draw $p_i \sim Beta(\alpha_i, \beta_i)$, then $Y_i \sim Bernoulli(p_i)$. Then $\alpha_{i+1} = \alpha_i + [Y_i = H] , \beta_{i+1} = \beta_i + [Y_i = T]$ .
      3. An urn contains $\alpha$ red balls and $\beta$ black balls. If a red/black ball is drawn, return 2 red/black balls to the urn. $Y$ is the number of observed red balls in n trials.

    • They are different from “For i from 1 to n, draw $p_i \sim Beta(\alpha, \beta)$, then $Y_i \sim Bernoulli(p_i)$”. This is just $Y \sim Bin(n, \alpha / (\alpha + \beta))$ because trials are independent.

    • Why [2] = [3]: $Pr_2(Y_i=H \mid history) = E[p_i \mid history] = \alpha_i / (\alpha_i + \beta_i)$

    • Why [1] = [2]:

      • In [1], $Y_i$’s are by definition n iid $Bernoulli(p)$.
        • [1] gives a joint distribution on $(p, Y_1 \dots Y_n)$ → we can discuss $\pi(p \mid Y_1 \dots Y_n)$
        • When we discuss $Pr_1(Y_i=H \mid history)$ below, p is by definition forced to be integrated out. We can add it back in using the law of total probability.

      $$ \begin{aligned} &Pr_1(Y_i=H \mid history) \\&= \int Pr_1(Y_i=H \mid p,history) \pi(p \mid history) dp \\&= E[Pr(Y_i=H \mid p,history) \mid history] = E[p \mid history].\end{aligned} $$

      • Note that $p \mid history \sim Beta(\alpha_i, \beta_i)$ in [2].
  • Def. (Multinomial) Multi-outcome version of Binomial. $f_{X}(x)=\frac{n!}{ \prod x_{i}!} \prod p_{i}^{x_{i}}$, where $\text{I\kern-0.15em E} X_{i}=np_{i}, VarX_{i}=np_{i}(1-p_{i})$.
  • Def. (Hypergeometric distribution) “The hypergeometric distribution is a discrete probability distribution that calculates the likelihood of k successes (s.t. the object drawn has a specified feature) in n draws when you are sampling from a small population consists of N objects with K objects with that feature without replacement”. $f_{X}(k)=\frac{ {K \choose k} {N-K \choose n-k} }{ N \choose n }$. $\text{I\kern-0.15em E} X=n N_{1}/N$, $Var X=n \frac{N_{1}}{N}\frac{N-K}{N}\frac{N-n}{N-1}$.
  • Def. (Bivariate Normal distribution) $\forall a, b \in \R, aX + bY$ is normal.