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# 1.S: Summary

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$$\newcommand\Vvarrho ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/01:_Fundamentals_of_Probability/1.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/01:_Fundamentals_of_Probability/1.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vvarsigma ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/01:_Fundamentals_of_Probability/1.S:_Summary), /content/body/p/span, line 1, column 23 $$
$$\newcommand\Vtau ParseError: invalid DekiScript (click for details) Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/01:_Fundamentals_of_Probability/1.S:_Summary), /content/body/p/span, line 1, column 23 $$
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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 $f(\By|\Bx)=\!\!\int\limits_0^1\!d\theta\,f(\By|\theta)\,\pi(\theta|\Bx)= {\SB(X+Y+\alpha,N-X+M-Y+\beta)\over\SB(X+\alpha,N-X+\beta)}\quad.$