Probability Theory Review

Preliminaries

Example Problem: Disease Diagnosis

  • [Question] Given a disease with prevalence of 1% and a test procedure with sensitivity ('true positive' rate) of 95% and specificity ('true negative' rate) of 85% , what is the chance that somebody who tests positive actually has the disease?
  • [Solution] Use probabilistic inference, to be discussed in this lecture.

Why Probability Theory?

  • Probability theory (PT) is the theory of optimal processing of incomplete information (see Cox theorem), and as such provides a quantitative framework for drawing conclusions from a finite (read: incomplete) data set.
  • Machine learning concerns drawing conclusions from (a finite set of) data and therefore PT is the optimal calculus for machine learning.
  • In general, nearly all interesting questions in machine learning can be stated in the following form (a conditional probability): $$p(\text{whatever-we-want-to-know}\, | \,\text{whatever-we-do-know})$$
    • For example:
      • Predictions $$p(\,\text{future-observations}\,|\,\text{past-observations}\,)$$
      • Classify a received data point $$p(\,x\text{-belongs-to-class-}k \,|\,x\,)$$
  • Information theory ("theory of log-probability") provides a source coding view on machine learning that is consistent with probability theory (more in part-2).

Probability Theory Notation

  • Define an event $A$ as a statement, whose truth is contemplated by a person, e.g.,
$$A = \text{'it will rain tomorrow'}$$
  • We write the denial of $A$, i.e. the event not-A, as $\bar{A}$.
probabilities
  • For any event $A$, with background knowledge $I$, the conditional probability of $A$ given $I$, is written as $$p(A|I)$$
  • The value of a probability is limited to $0 \le p(A|I) \le 1$.
  • All probabilities are in principle conditional probabilities of the type $p(A|I)$, since there is always some background knowledge.
  • Still, we often write $p(A)$ rather than $p(A|I)$ if the background knowledge $I$ is assumed to be obviously present. E.g., $p(A)$ rather than $p(\,A\,|\,\text{the-sun-comes-up-tomorrow}\,)$.
probabilities for random variable assignments
  • Note that, if $X$ is a random variable, then the assignment $X=x$ (with $x$ a value) can be interpreted as an event.
  • We often write $p(x)$ rather than $p(X=x)$ (hoping that the reader understands the context ;-)
  • In an apparent effort to further abuse notational conventions, $p(X)$ often denotes the full distribution over random variable $X$, i.e., the distribution for all assignments for $X$.
compound events
  • The joint probability that both $A$ and $B$ are true, given $I$ (a.k.a. conjunction) is written as $$p(A,B|I)$$
  • $A$ and $B$ are said to be independent, given $I$, if (and only if) $$p(A,B|I) = p(A|I)\,p(B|I)$$
  • The probability that either $A$ or $B$, or both $A$ and $B$, are true, given $I$ (a.k.a. disjunction) is written as $$p(A+B|I)$$

Probability Theory Calculus

  • Normalization. If you know that event $A$ given $I$ is true, then $p(A|I)=1$.
  • Product rule. The conjuction of two events $A$ and $B$ with given background $I$ is given by $$ p(A,B|I) = p(A|B,I)\,p(B|I) \,.$$
    • If $A$ and $B$ are independent given $I$, then $p(A|B,I) = p(A|I)$.
  • Sum rule. The disjunction for two events $A$ and $B$ given background $I$ is given by $$ p(A+B|I) = p(A|I) + p(B|I) - p(A,B|I)\,.$$
    • As a special case, it follows from the sum rule that $ p(A|I) + p(\bar{A}|I) = 1$
    • Note that the background information may not change, e.g., if $I^\prime \neq I$, then $$ p(A+B|I^\prime) \neq p(A|I) + p(B|I) - p(A,B|I)\,.$$
  • All legitimate probabilistic relations can be derived from the sum and product rules!
  • The product and sum rules are also known as the axioms of probability theory, but in fact, under some mild conditions, they can be derived as the unique rules for rational reasoning under uncertainty (Cox theorem, 1946).

Frequentist vs. Bayesian Interpretation of Probabilities

  • In the frequentist interpretation, $p(A)$ relates to the relative frequency that $A$ would occur under repeated execution of an experiment.
  • For instance, if the experiment is tossing a coin, then $p(\texttt{tail}) = 0.4$ means that in the limit of a large number of coin tosses, 40% of outcomes turn up as $\texttt{tail}$.
  • In the Bayesian interpretation, $p(A)$ reflects the degree of belief that event $A$ is true. I.o.w., the probability is associated with a state-of-knowledge (usually held by a person).
    • For instance, for the coin tossing experiment, $p(\texttt{tail}) = 0.4$ should be interpreted as the belief that there is a 40% chance that $\texttt{tail}$ comes up if the coin were tossed.
    • Under the Bayesian interpretation, PT calculus (sum and product rules) extends boolean logic to rational reasoning with uncertainty.
  • The Bayesian viewpoint is more generally applicable than the frequentist viewpoint, e.g., it is hard to apply the frequentist viewpoint to the event '$\texttt{it will rain tomorrow}$'.
  • The Bayesian viewpoint is clearly favored in the machine learning community. (In this class, we also strongly favor the Bayesian interpretation).

The Sum Rule and Marginalization

  • We discussed that every inference problem in PT can be evaluated through the sum and product rules. Next, we present two useful corollaries: (1) Marginalization and (2) Bayes rule
  • If $X$ and $Y$ are random variables over finite domains, than it follows from the sum rule that $$ p(X) = \sum_Y p(X,Y) = \sum_Y p(X|Y) p(Y) \,. $$
  • Note that this is just a generalized sum rule. In fact, Bishop (p.14) (and some other authors as well) calls this the sum rule.
    • EXERCISE: Proof the generalized sum rule.
  • Of course, in the continuous domain, the (generalized) sum rule becomes $$p(X)=\int p(X,Y) \,\mathrm{d}Y$$
  • Integrating $Y$ out of a joint distribution is called marginalization and the result $p(X)$ is sometimes referred to as the marginal probability.

The Product Rule and Bayes Rule

  • Consider 2 variables $D$ and $\theta$; it follows symmetry arguments that $$p(D,\theta)=p(D|\theta)p(\theta)=p(\theta|D)p(D)$$ and hence that $$ p(\theta|D) = \frac{p(D|\theta) p(\theta)}{p(D)}\,.$$
  • This formula is called Bayes rule (or Bayes theorem). While Bayes rule is always true, a particularly useful application occurs when $D$ refers to an observed data set and $\theta$ is set of model parameters that relates to the data. In that case,

    • the prior probability $p(\theta)$ represents our state-of-knowledge about proper values for $\theta$, before seeing the data $D$.
    • the posterior probability $p(\theta|D)$ represents our state-of-knowledge about $\theta$ after we have seen the data.

$\Rightarrow$ Bayes rule tells us how to update our knowledge about model parameters when facing new data. Hence,

Bayes rule is the fundamental rule for machine learning!

Bayes Rule Nomenclature

  • Some nomenclature associated with Bayes rule: $$ \underbrace{p(\theta | D)}_{\text{posterior}} = \frac{\overbrace{p(D|\theta)}^{\text{likelihood}} \times \overbrace{p(\theta)}^{\text{prior}}}{\underbrace{p(D)}_{\text{evidence}}} $$
  • Note that the evidence (a.k.a. marginal likelihood) can be computed from the numerator through marginalization since $$ p(D) = \int p(D,\theta) \,\mathrm{d}\theta = \int p(D|\theta)\,p(\theta) \,\mathrm{d}\theta$$
  • Hence, having access to likelihood and prior is sufficient to compute both the evidence and the posterior. To emphasize that point, Bayes rule is sometimes written as $$ p(\theta|D)\,p(D) = p(D|\theta)\, p(\theta)$$
  • For given $D$, the posterior probabilities of the parameters scale relatively against each other as $$ p(\theta|D) \propto p(D|\theta) p(\theta) $$

$\Longrightarrow$ All that we can learn from the observed data is contained in the likelihood function $p(D|\theta)$. This is called the likelihood principle.

The Likelihood Function vs the Sampling Distribution

  • Consider a model $p(D|\theta)$, where $D$ relates to variables that are observed (i.e., a "data set") and $\theta$ are model parameters.
  • In general, $p(D|\theta)$ is just a function of the two variables $D$ and $\theta$. We distinguish two interpretations of this function, depending on which variable is observed (or given by other means).
  • The sampling distribution (a.k.a. the data-generating distribution) $$p(D|\theta=\theta_0)$$ (which is a function of $D$ only) describes the probability distribution for data $D$, assuming that it is generated by the given model with parameters fixed at $\theta = \theta_0$.
  • In a machine learning context, often the data is observed, and $\theta$ is the free variable. For given observations $D=D_0$, the likelihood function (which is a function only of the model parameters $\theta$) is defined as $$\mathrm{L}(\theta) \triangleq p(D=D_0|\theta)$$
  • Note that $\mathrm{L}(\theta)$ is not a probability distribution for $\theta$ since in general $\sum_\theta \mathrm{L}(\theta) \neq 1$.
  • Technically, it is more correct to speak about the likelihood of a model (or model parameters) than about the likelihood of an observed data set. (Why?)

CODE EXAMPLE

Consider the following simple model for the outcome (head or tail) of a biased coin toss with parameter $\theta \in [0,1]$:

$$\begin{align*} y &\in \{0,1\} \\ p(y|\theta) &\triangleq \theta^y (1-\theta)^{1-y}\\ \end{align*}$$

We can plot both the sampling distribution (i.e. $p(y|\theta=0.8)$) and the likelihood function (i.e. $L(\theta) = p(y=0|\theta)$).

In [2]:
using Reactive, Interact, PyPlot
p(y,θ) = θ.^y .* (1 .- θ).^(1 .- y)
f = figure()
@manipulate for y=false, θ=0:0.1:1; withfig(f) do
        # Plot the sampling distribution
        subplot(221); stem([0,1], p([0,1],θ)); 
        title("Sampling distribution"); 
        xlim([-0.5,1.5]); ylim([0,1]); xlabel("y"); ylabel("p(y|θ=$(θ))");
        # Plot the likelihood function
         = range(0.0, stop=1.0, length=100)
        subplot(222); plot(, p(convert(Float64,y), )); 
        title("Likelihood function"); 
        xlabel("θ"); 
        ylabel("L(θ) = p(y=$(convert(Float64,y))|θ)");
    end
end
Out[2]:
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&#39;&gt;&lt;/img&gt;&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[]}]})</unsafe-script>"}}},"children":[{"props":{"id":"out","setInnerHtml":"<div 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The (discrete) sampling distribution is a valid probability distribution. However, the likelihood function $L(\theta)$ clearly isn't, since $\int_0^1 L(\theta) \mathrm{d}\theta \neq 1$.

Probabilistic Inference

  • Probabilistic inference refers to computing $$ p(\,\text{whatever-we-want-to-know}\, | \,\text{whatever-we-already-know}\,) $$
    • For example: $$\begin{align*} p(\,\text{Mr.S.-killed-Mrs.S.} \;&|\; \text{he-has-her-blood-on-his-shirt}\,) \\ p(\,\text{transmitted-codeword} \;&|\;\text{received-codeword}\,) \end{align*}$$
  • This can be accomplished by repeated application of sum and product rules.
  • For instance, consider a joint distribution $p(X,Y,Z)$. Assume we are interested in $p(X|Z)$: $$\begin{align*} p(X|Z) \stackrel{p}{=} \frac{p(X,Z)}{p(Z)} \stackrel{s}{=} \frac{\sum_Y p(X,Y,Z)}{\sum_{X,Y} p(X,Y,Z)} \,, \end{align*}$$ where the 's' and 'p' above the equality sign indicate whether the sum or product rule was used.
  • In the rest of this course, we'll encounter many long probabilistic derivations. For each manipulation, you should be able to associate an 's' (for sum rule), a 'p' (for product or Bayes rule) or an 'a' (for a model assumption) above any equality sign. If you can't do that, file a github issue :)

Working out the example problem: Disease Diagnosis

  • [Question] - Given a disease $D$ with prevalence of $1\%$ and a test procedure $T$ with sensitivity ('true positive' rate) of $95\%$ and specificity ('true negative' rate) of $85\%$, what is the chance that somebody who tests positive actually has the disease?
  • [Answer] - The given data are $p(D=1)=0.01$, $p(T=1|D=1)=0.95$ and $p(T=0|D=0)=0.85$. Then according to Bayes rule,
$$\begin{align*} p( D=1 &| T=1) \\ &= \frac{p(T=1|D=1)p(D=1)}{p(T=1)} \\ &= \frac{p(T=1|D=1)p(D=1)}{p(T=1|D=1)p(D=1)+p(T=1|D=0)p(D=0)} \\ &= \frac{0.95\times0.01}{0.95\times0.01 + 0.15\times0.99} = 0.0601 \end{align*}$$

Inference Exercise: Bag Counter

  • [Question] - A bag contains one ball, known to be either white or black. A white ball is put in, the bag is shaken, and a ball is drawn out, which proves to be white. What is now the chance of drawing a white ball?
  • [Answer] - Again, use Bayes and marginalization to arrive at $p(\text{white}|\text{data})=2/3$, see homework exercise

  • $\Rightarrow$ Note that probabilities describe a person's state of knowledge rather than a 'property of nature'.

  • [Excercise] - Is a speech signal a 'probabilistic' (random) or a deterministic signal?

Inference Exercise: Causality?

  • [Question] - A dark bag contains five red balls and seven green ones. (a) What is the probability of drawing a red ball on the first draw? Balls are not returned to the bag after each draw. (b) If you know that on the second draw the ball was a green one, what is now the probability of drawing a red ball on the first draw?
  • [Answer] - (a) $5/12$. (b) $5/11$, see homework.

  • $\Rightarrow$ Again, we conclude that conditional probabilities reflect implications for a state of knowledge rather than temporal causality.

PDF for the Sum of Two Variables

  • [Question] - Given two random independent variables $X$ and $Y$, with PDF's $p_x(x)$ and $p_y(y)$. What is the PDF of $$Z = X + Y\;?$$
  • [Answer] - Let $p_z(z)$ be the probability that $Z$ has value $z$. This occurs if $X$ has some value $x$ and at the same time $Y=z-x$, with joint probability $p_x(x)p_y(z-x)$. Since $x$ can be any value, we sum over all possible values for $x$ to get $$ p_z (z) = \int_{ - \infty }^\infty {p_x (x)p_y (z - x)\,\mathrm{d}{x}} $$
    • I.o.w., $p_z(z)$ is the convolution of $p_x$ and $p_y$.
  • Note that $p_z(z) \neq p_x(x) + p_y(y)\,$ !!
  • $\Rightarrow$ In linear stochastic systems theory, the Fourier Transform of a PDF (i.e., the characteristic function) plays an important computational role.
  • This list shows how these convolutions work out for a few common probability distributions.

CODE EXAMPLE

  • Consider the PDF of the sum of two independent Gaussians $X$ and $Y$:
$$\begin{align*} p_X(x) &= \mathcal{N}(\,x\,|\,\mu_X,\sigma_X^2\,) \\ p_Y(y) &= \mathcal{N}(\,y\,|\,\mu_Y,\sigma_Y^2\,) \\ Z &= X + Y \end{align*}$$
  • Performing the convolution (nice exercise) yields a Gaussian PDF for $Z$:
$$ p_Z(z) = \mathcal{N}(\,z\,|\,\mu_X+\mu_Y,\sigma_X^2+\sigma_Y^2\,). $$
In [1]:
using Reactive, Interact, PyPlot, Distributions
f = figure()
@manipulate for μx=0:0.1:4, σx=0.1:0.1:1.9,μy=0:0.1:4, σy=0.1:0.1:0.9; withfig(f) do
        μz = μx+μy; σz = sqrt(σx^2 + σy^2)
        x = Normal(μx, σx)
        y = Normal(μy, σy)
        z = Normal(μz, σz)
        range_min = minimum([μx-2*σx, μy-2*σy, μz-2*σz])
        range_max = maximum([μx+2*σx, μy+2*σy, μz+2*σz])
        range_grid = range(range_min, stop=range_max, length=100)
        plot(range_grid, pdf.(x,range_grid), "k-")
        plot(range_grid, pdf.(y,range_grid), "b-")
        plot(range_grid, pdf.(z,range_grid), "r-")
        legend([L"p_X", L"p_Y", L"p_Z"])
        grid()
    end
end
Out[1]:
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&#39;&gt;&lt;/img&gt;&quot;},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&#39;&gt;&lt;/img&gt;&quot;},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PDF for the Product of Two Variables

  • [Question] - Given two continuous random independent variables $X$ and $Y$, with PDF's $p_x(x)$ and $p_y(y)$. What is the PDF of $$Z = X Y \,\text{?}$$
  • Generally, this intergral does not lead to an analytical expression for $p_z(z)$. For example, the product of two independent variables that are both normally (Gaussian) distributed does not lead to a normal distribution.
  • Exception: the distribution of the product of two variables that both have log-normal distributions is again a lognormal distribution.
    • If $X$ has a normal distribution, then $Y=\exp(X)$ has a log-normal distribution.

Expectation and Variance

  • The expected value or mean is defined as $$\mathrm{E}[f] \triangleq \int f(x) \,p(x) \,\mathrm{d}{x}$$
  • The variance is defined as $$\mathrm{var}[f] \triangleq \mathrm{E} \left[(f(x)-\mathrm{E}[f(x)])^2 \right]$$
  • The covariance matrix between vectors $x$ and $y$ is defined as $$\begin{align*} \mathrm{cov}[x,y] &\triangleq \mathrm{E}\left[ (x-\mathrm{E}[x]) (y^T-\mathrm{E}[y^T]) \right]\\ &= \mathrm{E}[x y^T] - \mathrm{E}[x]\mathrm{E}[y^T] \end{align*}$$
    • Also useful as: $\mathrm{E}[xy^T] = \mathrm{cov}[x,y] + \mathrm{E}[x]\mathrm{E}[y^T]$

Example: Mean and Variance for the Sum of Two Variables

  • For any distribution of $x$ and $y$ and $z=x+y$,
\begin{align*} \mathrm{E}[z] &= \int_z z \left[\int_x p_x(x)p_y(z-x) \,\mathrm{d}{x} \right] \,\mathrm{d}{z} \\ &= \int_x p_x(x) \left[ \int_z z p_y(z-x)\,\mathrm{d}{z} \right] \,\mathrm{d}{x} \\ &= \int_x p_x(x) \left[ \int_{y^\prime} (y^\prime +x)p_y(y^\prime)\,\mathrm{d}{y^\prime} \right] \,\mathrm{d}{x} \notag\\ &= \int_x p_x(x) \left( \mathrm{E}[y]+x \right) \,\mathrm{d}{x} \notag\\ &= \mathrm{E}[x] + \mathrm{E}[y] \qquad \text{(always; follows from SRG-3a)} \end{align*}
  • Derive as an exercise that
\begin{align*} \mathrm{var}[z] &= \mathrm{var}[x] + \mathrm{var}[y] + 2\mathrm{cov}[x,y] \qquad \text{(always, see SRG-3b)} \notag\\ &= \mathrm{var}[x] + \mathrm{var}[y] \qquad \text{(if X and Y are independent)} \end{align*}

Linear Transformations

No matter how $x$ is distributed, we can easily derive that (do as exercise)

$$\begin{align} \mathrm{E}[Ax +b] &= A\mathrm{E}[x] + b \tag{SRG-3a}\\ \mathrm{cov}[Ax +b] &= A\,\mathrm{cov}[x]\,A^T \tag{SRG-3b} \end{align}$$
  • (The tag (SRG-3a) refers to the corresponding eqn number in Sam Roweis' Gaussian Identities notes.)

Review Probability Theory

  • Interpretation as a degree of belief, i.e. a state-of-knowledge, not as a property of nature.
  • We can do everything with only the sum rule and the product rule. In practice, Bayes rule and marginalization are often very useful for computing
$$p(\,\text{what-we-want-to-know}\,|\,\text{what-we-already-know}\,)\,.$$
  • Bayes rule $$ p(\theta|D) = \frac{p(D|\theta)p(\theta)} {p(D)} $$ is the fundamental rule for learning!
  • That's really about all you need to know about probability theory, but you need to really know it, so do the exercises.
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