1.S: Summary

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References

• C. Gardiner, Stochastic Methods ($4^{th}$ edition, Springer-Verlag, 2010) Very clear and complete text on stochastic methods with many applications.
• J. M. Bernardo and A. F. M. Smith, Bayesian Theory (Wiley, 2000) A thorough textbook on Bayesian methods.
• D. Williams, Weighing the Odds: A Course in Probability and Statistics (Cambridge, 2001) A good overall statistics textbook, according to a mathematician colleague.
• E. T. Jaynes, Probability Theory (Cambridge, 2007) An extensive, descriptive, and highly opinionated presentation, with a strongly Bayesian approach.
• A. N. Kolmogorov, Foundations of the Theory of Probability (Chelsea, 1956) The Urtext of mathematical probability theory.

Summary

$\bullet$ Discrete distributions: Let $n$ label the distinct possible outcomes of a discrete random process, and let $p\ns_n$ be the probability for outcome $n$. Let $A$ be a quantity which takes values which depend on $n$, with $A\ns_n$ being the value of $A$ under the outcome $n$. Then the expected value of $A$ is $\langle A \rangle = \sum_n p\ns_n\,A\ns_n$, where the sum is over all possible allowed values of $n$. We must have that the distribution is normalized, $\langle 1 \rangle = \sum_n p\ns_n=1$.

$\bullet$ Continuous distributions: When the random variable $\Bvphi$ takes a continuum of values, we define the probability density $P(\Bvphi)$ to be such that $P(\Bvphi)\,d\mu$ is the probability for the outcome to lie within a differential volume $d\mu$ of $\Bvphi$, where $d\mu = W(\Bvphi)\prod_{i=1}^n d\vphi\ns_i$, were $\Bvphi$ is an $n$-component vector in the configuration space $\ROmega$, and where the function $W(\Bvphi)$ accounts for the possibility of different configuration space measures. Then if $A(\Bvphi)$ is any function on $\ROmega$, the expected value of $A$ is $\langle A\rangle=\int\limits_\ROmega\!d\mu\>P(\Bvphi)\,A(\Bvphi)$.

$\bullet$ Central limit theorem: If $\{x\ns_1,\ldots,x\ns_N\}$ are each independently distributed according to $P(x)$, then the distribution of the sum $X=\sum_{i=1}^N x\ns_i$ is $$\CP\ns_N(X)=\!\!\impi dx\ns_1\cdots\!\!\impi dx\ns_N\,P(x\ns_1)\cdots P(x\ns_N)\> \delta\Big(X-\sum_{i=1}^N x\ns_i\Big)=\!\!\impi {dk\over 2\pi}\> \left[ \HP(k) \right]^N \! e^{ikX}\ ,$$ where $\HP(k)=\int\!dx\>P(x)\,e^{-ikx}$ is the Fourier transform of $P(x)$. Assuming that the lowest moments of $P(x)$ exist, $\ln\!\big[\HP(k)\big]=-i\mu k -\half\sigma^2 k^2 + \CO(k^3)$, where $\mu=\langle x\rangle$ and $\sigma^2=\langle x^2\rangle - \langle x\rangle^2$ are the mean and standard deviation. Then for $N\to\infty$, $$P\ns_N(X)=(2\pi N\sigma^2)^{-1/2}\,e^{-(X-N\mu)^2/ 2N\sigma^2}\ ,$$ which is a Gaussian with mean $\langle X\rangle = N\mu$ and standard deviation $\sqrt{\langle X^2\rangle - \langle X\rangle^2}=\sqrt{N}\,\sigma$. Thus, $X$ is distributed as a Gaussian, even if $P(x)$ is not a Gaussian itself.

$\bullet$ Entropy: The entropy of a statistical distribution is $\{p\ns_n\}$ is $S=-\sum_n p\ns_n \ln p\ns_n$. (Sometimes the base 2 logarithm is used, in which case the entropy is measured in bits.) This has the interpretation of the information content per element of a random sequence.

$\bullet$ Distributions from maximum entropy: Given a distribution $\{p\ns_n\}$ subject to $(K+1)$ constraints of the form $\CX^a=\sum_n X^a_n \, p\ns_n$ with $a\in\{0,\ldots,K\}$, where $\CX^0=X^0_n=1$ (normalization), the distribution consistent with these constraints which maximizes the entropy function is obtained by extremizing the multivariable function $$S^*\big(\{p\ns_n\},\{\lambda\ns_a\}\big)=-\sum_n p\ns_n \ln p\ns_n - \sum_{a=0}^K \lambda\ns_a \Big(\sum_n X^a_n\,p\ns_n - \CX^a\Big) \ ,$$ with respect to the probabilities $\{p\ns_n\}$ and the Lagrange multipliers $\{\lambda\ns_a\}$. This results in a Gibbs distribution, $$p\ns_n={1\over Z}\exp\!\left\{-\sum_{a=1}^K \lambda\ns_a X^a_n\right\}\ ,$$ where $Z=e^{1+\lambda\ns_0}$ is determined by normalization, $\sum_n p\ns_n = 1$ ( the $a=0$ constraint) and the $K$ remaining multipliers determined by the $K$ additional constraints.

$\bullet$ Multidimensional Gaussian integral: $$\impi dx\ns_1\cdots\!\impi dx\ns_n \> \exp\Big(\!-\half \, x\ns_i \, A\ns_{ij} \, x\ns_j + b\ns_i\,x\ns_i\Big)= \bigg({(2\pi)^n\over \det\!A}\bigg)^{\!1/2}\exp\Big(\half \, b\ns_i\,A^{-1}_{ij}\,b\ns_j\Big)\ .$$

$\bullet$ Bayes’ theorem: Let the conditional probability for $B$ given $A$ be $P(B|A)$. Then Bayes’ theorem says $P(A|B)=P(A)\cdot P(B|A) \, / \, P(B)$. If the ’event space’ is partitioned as $\{A\ns_i\}$, then we have the extended form, $$P(A\ns_i|B)={P(B|A\ns_i)\cdot P(A\ns_i)\over\sum_j P(B|A\ns_j)\cdot P(A\ns_j)}\ .$$ When the event space is a ‘binary partition’ $\{A,\neg A\}$, as is often the case in fields like epidemiology ( test positive or test negative), we have $$P(A|B)={P(B|A)\cdot P(A)\over P(B|A)\cdot P(A) + P(B|\neg A)\cdot P(\neg A)}\ . \label{Bayesbinary}$$ Note that $P(A|B)+P(\neg A|B)=1$ (which follows from $\neg\neg A = A$).

$\bullet$ Updating Bayesian priors: Given data in the form of observed values $\Bx=\{x\ns_1,\ldots,x\ns_N\}\in\CX$ and a hypothesis in the form of parameters $\Btheta=\{\theta\ns_1,\ldots,\theta\ns_K\}\in\Theta$, we write the conditional probability (density) for observing $\Bx$ given $\Btheta$ as $f(\Bx|\Btheta)$. Bayes’ theorem says that the corresponding distribution $\pi(\Btheta|\Bx)$ for $\Btheta$ conditioned on $\Bx$ is $$\pi(\Btheta|\Bx)={f(\Bx|\Btheta)\,\pi(\Btheta)\over\int\limits_\Theta\!d\Btheta'\> f(\Bx|\Btheta')\,\pi(\Btheta')}\ ,$$ We call $\pi(\Btheta)$ the prior for $\Btheta$, $f(\Bx|\Btheta)$ the likelihood of $\Bx$ given $\Btheta$, and $\pi(\Btheta|\Bx)$ the posterior for $\Btheta$ given $\Bx$. We can use the posterior to find the distribution of new data points $\By$, called the posterior predictive distribution, $f(\By|\Bx)=\int\limits_\Theta\!d\Btheta\,f(\By|\Btheta)\,\pi(\Btheta|\Bx)\,.$ This is the update of the prior predictive distribution, $f(\Bx)=\int\limits_\Theta\!d\Btheta\,f(\Bx|\Btheta)\,\pi(\Btheta)$ . As an example, consider coin flipping with $f(\Bx|\Btheta)=\theta^X\,(1-\theta)^{N-X}$, where $N$ is the number of flips, and $X=\sum_{j=1}^N x\ns_j$ with $x\ns_j$ a discrete variable which is $0$ for tails and $1$ for heads. The parameter $\theta\in[0,1]$ is the probability to flip heads. We choose a prior $\pi(\theta)=\theta^{\alpha-1}\,(1-\theta)^{\beta-1}/\SB(\alpha,\beta)$ where $\SB(\alpha,\beta)=\RGamma(\alpha)\,\RGamma(\beta)/\RGamma(\alpha+\beta)$ is the Beta distribution. This results in a normalized prior $\int\limits_0^1 \!d\theta\,\pi(\theta)=1$. The posterior distribution for $\theta$ is then $$\pi(\theta|x\ns_1,\ldots,x\ns_N)={f(x\ns_1,\ldots,x\ns_N|\theta)\,\pi(\theta)\over\int_0^1\!d\theta'\, f(x\ns_1,\ldots,x\ns_N|\theta')\,\pi(\theta')} ={\theta^{X+\alpha-1} (1-\theta)^{N-X+\beta-1}\over\SB(X+\alpha,N-X+\beta)}\ .$$ The prior predictive is $f(\Bx)=\int\limits_0^1\!d\theta f(\Bx|\theta)\,\pi(\theta)=\SB(X+\alpha,N-X+\beta)/\SB(\alpha,\beta)$ , and the posterior predictive for the total number of heads $Y$ in $M$ flips is $$f(\By|\Bx)=\!\!\int\limits_0^1\!d\theta\,f(\By|\theta)\,\pi(\theta|\Bx)= {\SB(X+Y+\alpha,N-X+M-Y+\beta)\over\SB(X+\alpha,N-X+\beta)}\quad.$$