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begin{document}
small
begin{multicols*}{5}
section{probability}

subsection*{Random Variables}
Joint.p.d.$p(left.vec{X})=p(x,y)right|sumsum p(x,y)=1$.
Margin/Law.of.Total.P.(eliminates$y$):
$p(x,y)=sum_z p(x,y,z)$.
%$p(x)=underset{y}{Sigma}p(x,y),p(x,y)=underset{z}{Sigma}p(x,y,z)$.
Conditional.prob.from.joint.
$left.p(x|y=y_1)right|int p(x|y=y_1)dx=1 rightarrow
frac{p(x,y=y_1)}{p(y=y_1)} = frac{p(x,y=y_1)}{int p(x,y=y_1)dx}=frac{p(x,y_1}{p(y_1}$.
$p(x,y) = p(x)p(y|x)=p(y)p(x|y)$.Generalize.
$p(x_1,x_2,cdots,x_n)=p(x_1)p(x_2|x_1)p(x_3|x_1,x_2)cdots p(x_n|x_1,cdots,x_{n-1})$.
Bayes.$p(y|x)=frac{p(y)p(x|y)}{p(x)}=frac{p(y)p(x|y)}{sum_y p(x,y)}=frac{p(y)p(x|y)}{sum_y p(y)p(x|y)}$\
subsection*{Independence}
Indep.=Marginal.indep:$p(x|y)=p(x)$($rightarrow p(y|x)=p(y)$).$leftrightarrow$.Joint.factorizes. $p(x,y)=p(x)p(y)$.
$x_1,x_2,cdots,x_n$.i.i.d$leftrightarrow$.they.are.indep.and.$p(x_1)=cdots=p(x_n)$.
Conditional.indep.given$x_3$:$p(x_1,x_2|x_3)=p(x_1|x_3)p(x_2|x_3)$.indep.$centernot rightarrow$cond.indep.$centernot rightarrow$.indep.\(Not.knowing.q.then.coin.tosses.not.indep.).
subsection*{Expectation}
Expectation.(generalized.average.with.prob.distr)
$E[f(x)]=sum f(x)p(x)dx,E[f(x,y)]=iint f(x,y) p(x,y) dx dy $
$f=x$:mean.$mu_x$.$f=x-mu_x$:var.$f=(x-mu_x)(y-mu_y)$:cov(x,y).
affine:$E[af+b]=aE[f]+b$.$text{cov}(x,y)=E(xy)-E(x)E(y)$.Convex$g$.$g(E[X])leq E[g(X)]$.indep.x.y.$E[f(x)g(y)]=E[f(x)]E[g(y)]rightarrow text{cov}(x,y)=0$.Cond.exp.$E[f(x,y)|y]=E_{p(x|y)}[f(x,y)]=sum_xf(x,y)p(x|y)$
subsection*{Common distributions}
Bernoulli.$x in {0,1},lambda in [0,1]: p(x) = lambda^x (1-lambda)^{1-x}=text{Ber}(x|lambda)$. Beta.$
left.p(lambda)=frac{1}{B(alpha,beta)}lambda^{alpha-1}(1-lambda)^{beta-1}=text{Beta}(lambda|alpha,beta)
right| alpha > 0,beta > 0.
left|B(alpha,beta) = frac{Gamma(alpha)Gamma(beta)}{Gamma(alpha+beta)}right|Gamma(z)=int_0^infty t^{z-1}e^{-t}dt.
E(lambda)=frac{alpha}{alpha+beta}
$.Cat.$
x in {0:K},lambda_k in [0,1]:
sum_{k=1}^K p(x=k)= sum lambda_k = 1.
p( textbf{x} = [textbf{e}]_k) = prod_{j=1}^K lambda_j^{e_j} = lambda_k =text{Cat}(textbf{x}|lambda)$. Dirichlet.$p(lambda)=
frac{1}{B(alpha)}prod_{k=1}^K lambda_k^{alpha_k-1} = text{Dir} (lambda|alpha)$.Normal$
p(x)=frac{1}{sqrt{2pisigma^2}}exp[-frac{(x-mu)^2}{sigma^2}]$.NInvGamma.MVN(is.a.joint)$
p(textbf{x})=frac{1}{(2pi)^{d/2}|Sigma|^{1/2}}
exp[-frac{1}{2}(textbf{x} - mu)^T Sigma^{-1} (textbf{x} - mu )]
= N(textbf{x}|mu,Sigma)$.
Covar $Sigma succ 0, Sigma_{ii}=text{Var}(x_i),Sigma_{ij}=text{Cov}(x_i,x_j)$, Precision $Lambda = Sigma^{-1}$.NInvWishart.
% Usually x_i are all i.i.d that's why cov(x_i,xj) looks nice eventhough there is p(x_i,x_j) in it
% formally cov and var defined through p, thus so is Sigma
subsection*{Conjugate distributions}
E.g.$text{Ber}(x|lambda)text{Beta}(lambda|alpha,beta) = C text{Beta}(lambda|x + alpha,1-x+beta) rightarrow$.Bayes:if.prior.conjugate.likelihood.then.posterior=shifted.prior.And.p(x).cancels.C
estimation.using.posterior(likeMAP).ez.Prediction.$
p(x^*|x)=int p(x^*,lambda|x)dlambda=
int p(x^*|lambda,x)p(lambda|x)dlambda=
int p(x^*|lambda)p(lambda|x)dlambda$Conjugate!

end{multicols*}
end{document}



subsection*{Naive-Bayes Classifiers}