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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

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.

Special Distributions

Beta distribution $B(r,s)$
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]
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. $f_{Y}(k)={n \choose k} p^{k} (1-p)^{n-k},k=1, 2, \dots, n$. Binomial process is that $Y_n$ to be the number of successes in the first n $Be(p)$ trials.
- 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 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”. $f_{X}(k)=\frac{ {K \choose k} {N-K \choose n-k} }{ N \choose n }$. $\text{I\kern-0.15em E} X=n N_{1}/N$, $Var X=n \frac{N_{1}}{N}\frac{N-K}{N}\frac{N-n}{N-1}$.

Sampling

“Probabilistic computations are approximated by replacing probability distributions with samples from those distributions.”

Simple Monte Carlo

$\mu:=E_{p(x)}[f(x)]:= \int f(x) p(x)dx \approx \frac1m \sum f(x^{(i)}):=\hat \mu_{MC}$
Analysis
- $\hat \mu_{MC}$ is consistent: $(\hat \mu_{MC} - \mu) \to 0$ almost surely as $m \to \infty$
- $\hat \mu_{MC}$ is unbiased: $E[\hat \mu_{MC}]=\mu$
- $\hat \mu_{MC}$ is asymptotically normal: $\sqrt m (\hat \mu_{MC}-\mu) \to N(0,\sigma^2_{MC})$ in distribution, where $\sigma^2_{MC}=E_{p(x)}[(f(x)-\mu)^2]$ the variance.
“The method of choice in cases where it is straightforward to sample from p(x) and the fluctuations in f(x) are not correlated with p(x).”