Chapter 7 Probability

7.1 Bayesian Inference

The Bayesian model for inference adds to the statistical model \(\{f_\theta : \theta \in \Omega\}\) a prior probability measure \(\Pi\) for \(\theta\). For example, suppose \(\theta\) is the probability a flipping a coin to heads, with \(\Omega = [0,1]\). The statistician’s conservative believe \(\pi\) may be a wide bell curve centered on \(\theta=0.5\), or maybe the statistician is quite confident, so \(\pi\) is a daring narrow bell curve around \(\theta=0.5\). By the law of total probability, the probability of event \(s\) is a joint distribution, \(\pi(\theta)f_\theta(s)\). The marginal distribution of \(s\) is \(m(s) = \int_\Omega \pi(\theta)f_\theta(s)d\theta.\)

After \(s\) is observed the posterior distribution of \(\theta\) is the conditional distribution of \(\theta\) given \(s\) is

\[\pi(\theta|s) = \frac{\pi(\theta)f_\theta(s)}{m(s)}\]

Suppose you observe a sample from a Bernoulli distribution with an unknown success probability, \(\theta \in [0,1].\) You take a uniform prior, \(\pi= Beta(\alpha, \beta) = Beta(1,1).\). Your sample of \(n = 40\) includes \(n\bar{x} = 10\) successes. Then the posterior of \(\theta\) is \(Beta(11,31):\)

The likelihood \(\theta^{n\bar{x}}\left(1-\theta\right)^{n(1-\bar{x})}\) times the prior \(B^{-1}(\alpha, \beta)\theta^{\alpha-1}(1-\theta)^{\beta-1}.\)

7.2 Stochastic Processes

Stochastic processes are proceed randomly over time.

The simplest version is the random walk. A random walk is a sequence \(\{X_n\}\) of random variables, with \(X_0\) = 1 and \(P\left(X_{n+1} = X_n + 1 \right) = p\) where \(p\) is the probability of “success” and its complement is \(q= 1-p\). It follows that, given an initial value \(a\), \(P\left(X_n = a + k\right) = \binom{n}{\frac{n+k}{2}} p^{(n+k)/2}q^{(n-k)/2}\) for \(k = -n, -n+2, -n+4, \dots,n\), and \(E\left(X_n\right) = a + n(2p-1)\).

7.2.1 Markov Chains

A Markov chain is the random motion of an object. Given a state space \(S\) of all places an object can go, and a set of transition probabilities \(\{p_{ij}: i,j \in S\}\) to move from state \(i\) to \(j\), and a probability \(u_i = P\left(X_0 = i\right)\) that the object starts at state \(i\), then \(P\left(X_{n+1} = j|X_n =i\right)=p_{ij}\)

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