Week 3: probability and the ace of Bayes
Lecture 3.1: the Bayes Theorem
Lecture 3.1: getting to know the world; probability in cognition; the Ace of Bayes
- How the brain can know the world
- A digression
- The fundamental role of probability in cognition
- The Bayes Theorem [including a proof, which also appears in
Appendix A in the book] and the subjective approach to probability
- Putting the Bayes Theorem to work
the brain's predicament
the brain's predicament: inside of a dog... wait, what?
the brain's predicament: inside of a dog skull
the brain's predicament (cont.)
How and in what sense can the brain get to KNOW the world?
The control of behavior requires that the brain
perform MEASUREMENTS on the outside world.
Think of this is as intelligence-gathering for the sake of the
command-and-control processes that reside in the
War Room.
Any system trying to get to know the world through
measurements must deal with UNCERTAINTY.
Is it OK to use WAR language in everyday life? [as in "WAR ON
CLIMATE CHANGE"]
[AN ASIDE] what is war good for?
cui bono? ["who profits?"]
[AN ASIDE] what who is war good for?
cui bono? ["who profits?"]
[AN ASIDE] what who is war good for?
[from The Intercept_]
A war between China and Taiwan will be extremely good for
business at America’s Frontier Fund, a tech investment outfit
whose co-founder and CEO sits on both the State Department
Foreign Affairs Policy Board and President Joe Biden’s
Intelligence Advisory Board, according to audio from a February
1 event.
Gilman Louie,
AFF’s
co-founder and current CEO, serves as chair of the National
Intelligence University, advises Biden through his
Intelligence Advisory Board, and was
tapped for the State Department’s Foreign
Affairs Policy Board by Secretary of State Antony Blinken
in 2022. Louie previously ran
In-Q-Tel, the
CIA’s venture capital arm.
[AN ASIDE] what who is war good for?
"The key question that should be asked about any war is: who
benefits from it? [...] The answer to this question is
inconvenient, but hard to argue with: it is the power elites that
reap a lion’s share of the material gain from war and that avoid
most of its human toll; and it is the ordinary people on all sides
who foot most of the bill. Perhaps instead of shyly thanking
random military personnel at airports for their service, the
children whom we encourage to do that should be taught to ask
“What made you enlist?” and “Do you realize whose cause you
serve?”"
— Life, Death, and Other Inconvenient
Truths (MIT Press, 2020), chapter 37: War.
BACK TO the brain's predicament: the role of probability and statistics in cognition
How and in what sense can the brain get to KNOW the world?
Any system trying to get to know the world through measurements must
deal with UNCERTAINTY. Therefore, the following observation is
crucially important:
"All knowledge resolves itself into probability"
David Hume
A Treatise of Human Nature (1740)
Probability theory is NOT about capitulating in
the face of uncertainty: it quantifies uncertainty and makes it formally
manageable.
on the importance of statistical data and methods
The joint probability distribution
$$
p(X,Y)
$$
is the most that can be known about
\(X\) and \(Y\) through observation.
[If you are allowed to intervene, you can
learn more, by doing science.]
You can estimate \(p(X,Y)\) by dividing the range of \(X\) and of \(Y\) into bins and
counting items that fall within each bin.
[Think of the values of \(X\) coding apple color; \(Y\) coding apple crunchiness.]
conditional probability, darts, and Venn diagrams
\(\require{color}\)
As an example, the conditional probability of putting a dart into that part of the inside of the big \(\textrm{O}\)
which is
\({\color{red} ♡}\)-colored
is defined as
$$
P(\textrm{O} \mid {\color{red} ♡}) = \frac{P(\textrm{O},{\color{red} ♡})}{P({\color{red} ♡})}
$$
-
Intuition: the conditional \(P(\textrm{O} \mid {\color{red} ♡})\)
is larger when the joint (intersection) \(P(\textrm{O},{\color{red}
♡})\) is larger; and smaller when the marginal \(P({\color{red}
♡})\) is larger (because then there are more ways of landing within
\({\color{red} ♡}\) but not within \(\textrm{O}\)).
-
An application: if you know how often apples are both OMGtasty and
\({\color{red} red}\),
\(P(OMGtasty,{\color{red} red})\), you can calculate the conditional probability of an apple
being OMGtasty, given that it is \({\color{red} red}\): by
definition,
$$
P(OMGtasty \mid {\color{red} red}) = \frac{P(OMGtasty,{\color{red}
red})}{P({\color{red} red})}
$$
using conditional probability in data-driven learning and generalization
The computational essence of categorization and regression:*
Classification
1. estimate the probability of each possible class label, given the values
of the object's features:
$$
p({\cal C}_i \mid x_1, x_2)
$$
2. choose
the class with the largest probability.
Example: categorization (given \(size\) and \(color\), predict \(crunchy/mushy\)).
Regression
1. estimate the probability of each possible output value,
given the input value(s):
$$
p(y \mid x)
$$
2. choose the output value
with the largest probability.
Example: estimation (given \(color\), predict \(HOW tasty\)); also
visual-motor coordination.
*
NOTE: input/output or stimulus-response mapping (which includes categorization and regression)
by no means covers everything that minds do to control behavior,
but it is an indispensable conceptual starting point.
probability estimation / generalization as statistical inference & decision-making
The computational essence of categorization and regression:
— Both classification and regression are underdetermined and
therefore must rely on extra assumptions (as in regularization; more about this next week).
— Both require probability estimation.
the kind of probability estimation required for learning and generalization
Continuing the example of learning to deal with apples:
\({\cal C}_1\) = "crunchy apple"
\({\cal C}_2\) = "mushy apple"
\(x_1\) : color dimension
\(x_2\) : size dimension
There is a bit of a problem.
Suppose that we're looking at an apple \(A\)
that has color \(x_1^{(A)}\) and size \(x_2^{(A)}\).
| from experience, we may know |
\(p\left(x_1^{(Z)}, x_2^{(Z)} \mid {\cal C}_1\right)\) |
— how often the crunchy apples \(Z\) we tasted happened to be of
a particular color and size |
| but what we need to know is |
\(p\left({\cal C}_1 \mid x_1^{(A)}, x_2^{(A)}\right)\) |
— how likely an apple \(A\) of this color and
size is to be crunchy (before tasting it) |
the kind of probability estimation required for learning
HELP!!!
is on the way: the Bayes Theorem.
the Bayes Theorem follows immediately from the definition of conditional probability
The
conditional probability of
B given
A
(think darts) is defined as the ratio of two areas:
$$
p(B \mid A) = \left\vert A \& B\right\vert / \left\vert A\right\vert
$$
On the right, divide numerator and denominator by the area of the
"universe" \(\left\vert U\right\vert\) to obtain a ratio of probabilities:
$$
p(B \mid A) = p(A \& B) / p(A)
$$
Now, by the definition of
conditional probability,
the
joint probability, which depends symmetrically on \(A\)
and \(B\), can be expressed in two equivalent ways:
$$
\begin{align}
p(A \& B) &= p(A) p(B \mid A) = \\
&= p(B) p(A \mid B) = p(B \& A)
\end{align}
$$
Suppose
\(B\) is a hypothesis ("apple is
crunchy"), and
\(A\)
is data ("apple is red"). We can now
estimate the
probability of the
hypothesis being true, given the
data:
$$
p({\color{gray} B} \mid \mathbf{A}) = \frac{p(\mathbf{A} \mid
{\color{gray} B}) p({\color{gray} B})}{p(\mathbf{A})}
$$
an application: Bayes in wireframe shape perception [h/t Dan Kersten]
 |
 |
|
Must find the probabilities of various conceivable shape
interpretations (hypotheses),
given the image (data). According to Bayes, $$p(S\mid I) \propto p(I\mid S)p(S)$$
|
The likelihood term, \(p(I\mid S)\), rules out shapes \(S\)
that are inconsistent with the image \(I\) (here, spheres,
cones, etc.).
|
Bayes in wireframe shape perception (cont.)
the big picture: the probabilistic approach to cognition (Chater et al., 2006)
"The brain is an information processor; and information processing
typically involves inferring new information from information that
has been derived from the senses, from linguistic input, or from
memory. This process of inference from old to new is,
outside pure mathematics, typically uncertain."
"Probability theory is, in essence, a calculus for uncertain
inference, according to
the SUBJECTIVE INTERPRETATION OF PROBABILITY.
Thus probabilistic methods have potentially broad application to
uncertain inferences:
— from sensory input to environmental layout;
— from speech signal to semantic interpretation;
— from goals to motor output;
— or from observations and experiments to regularities in nature."
subjective probability (Chater et al., 2006)
"Crucially, the frequency interpretation of probability is not in
play here — in cognitive science applications, probabilities refer
to 'DEGREES OF BELIEF'.
Thus, a person's degree of belief that a coin that has rolled
under the table has come up heads might be around 1/2; this degree
of belief might well increase rapidly to 1 as she moves her head,
bringing the coin into view. Her friend, observing the same
event, might have
different prior assumptions and obtain a different stream of sensory
evidence.
Thus the two people are viewing the same event, but their belief
states and hence their subjective probabilities might
differ. Moreover, the relevant information is defined by the
specific details of the situation. This particular pattern of
prior information and evidence will
never be repeated, and hence cannot define a limiting
frequency."
working with subjective probabilities (Chater et al., 2006)
"The subjective interpretation of probability generally aims to
evaluate CONDITIONAL PROBABILITIES, \(Pr(h_j\mid d)\), that is,
probabilities of alternative hypotheses, \(h_j\) (about the state of
reality), given certain data, \(d\) (e.g. available to the
senses). By Bayes' theorem,
$$
Pr(h_j \mid d) = \frac{Pr(d \mid h_j) Pr(h_j)}{Pr(d)}
$$
The centrality of Bayes' Theorem to the subjective approach to
probability has led to the approach commonly being known as the
Bayesian approach. But the real content of the approach is the
subjective interpretation of probability; Bayes' Theorem itself is just an
elementary, if spectacularly productive, identity in probability
theory."
probability models are useful on many levels (Chater et al., 2006)
"Sophisticated probabilistic models can be related to cognitive processes
in a variety of ways. This variety can usefully be understood in terms of
Marr's celebrated distinction between three levels of computational
explanation: the computational level, which specifies the nature of the
cognitive problem being solved, the information involved in solving it,
and the logic by which it can be solved; the algorithmic level, which
specifies the representations and processes by which solutions to the
problem are computed; and the implementational level, which specifies how
these representations and processes are realized in neural terms.
Finally, turning to the implementational level, one may
ask whether THE BRAIN ITSELF SHOULD BE VIEWED IN PROBABILISTIC
TERMS. Intriguingly, many of the sophisticated probabilistic models that
have been developed with cognitive processes in mind map naturally onto
highly distributed, autonomous, and parallel computational architectures,
which seem to capture the qualitative features of neural
architecture."
basic Bayes: how to use the estimated posterior? (Griffiths & Yuille, 2006)
Assume that we have an agent who is attempting to infer the process that
was responsible for generating some data, \(d\). Let \(h\) be a hypothesis about
this process, and \(P(h)\) — the prior probability that the agent would have
accepted \(h\) before seeing \(d\). How should the agent's beliefs change
in the light of the evidence provided by \(d\)? To answer this question, we
need a procedure for computing the posterior probability, \(P(h
\mid d)\). This is provided by the Bayes Theorem:
$$
P(h \mid d) = \frac{P(d \mid h) P(h)}{P(d)}
$$
How can the posterior be used to guide action?
The denominator is obtained by summing over [the mutually exclusive]
hypotheses, a procedure known as marginalization:
$$
P(d) = \sum_{h^{\prime}\in H} P(d \mid h^{\prime}) P(h^{\prime})
$$
where \(H\) is the set of all hypotheses considered by the agent.
using the posterior
How can the posterior DISTRIBUTION be used to guide action?
If this is the posterior \(P(h \mid d)\) for a certain regression
problem involving data \(d\), what value of \(h\) would you
choose as the answer to the problem?
using the posterior: Bayesian decision & control (Griffiths & Yuille, 2006, Box 1)
Bayesian decision theory introduces a loss function \(L\left(h,
\alpha\left(d\right)\right)\) for the cost of making a decision \(\alpha(d)\) when the
input is \(d\) and the true hypothesis [true state of affairs] is \(h\). It proposes selecting the
decision function or rule \(\alpha^{\star}(\cdot)\) that minimizes the RISK, or
EXPECTED LOSS:
$$
R(\alpha) = \sum_{h,d} L\left(h, \alpha\left(d\right)\right) P(h, d)
$$
or in words: weight the loss for each possible combination of
data (which dictates a decision) and true hypothesis by how probable that
combination is, and sum these resulting values. This is RATIONAL DECISION MAKING.
In classification, \(L\) can be chosen so that the same penalty is paid for all
wrong decisions:
\(L\left(h, \alpha\left(d\right)\right) = 1\) if \(\alpha\left(d\right) \neq h\)
and
\(L\left(h, \alpha\left(d\right)\right) = 0\) if \(\alpha\left(d\right) = h\).
Then the best decision rule
is the maximum a posteriori (MAP) estimator
\(\alpha^{\star}(d) = \textrm{argmax}_{h} P(h \mid d)\).
using the posterior: Bayesian decision & control (Griffiths & Yuille, 2006, Box 1)
Bayesian decision theory introduces a loss function \(L\left(h,
\alpha\left(d\right)\right)\) for the cost of making a decision \(\alpha(d)\) when the
input is \(d\) and the true hypothesis [true state of affairs] is \(h\). It proposes selecting the
decision function or rule \(\alpha^{\star}(\cdot)\) that minimizes the RISK, or
EXPECTED LOSS:
$$
R(\alpha) = \sum_{h,d} L\left(h, \alpha\left(d\right)\right) P(h, d)
$$
or in words: weight the loss for each possible combination of
data (which dictates a decision) and true hypothesis by how probable that
combination is, and sum these resulting values. This is RATIONAL DECISION MAKING.
In regression, the loss function can take the form of the square of the error:
\(L\left(h, \alpha\left(d\right)\right) = \left\{h−\alpha\left(d\right)\right\}^2\)
Then the best solution is the
posterior mean, that is, the probabilistically weighted average
of all possible (numerical in this case) hypotheses: \(\sum_{h} h P(h \mid d)\).
generative vs. empirical risk minimization approaches
An important distinction: generative models vs. empirical risk minimization approaches —
In many situations, we will not know the distribution \(P(h, d)\) exactly
but will instead have a set of labelled samples \(\left\{\left(h_i, d_i\right) : i =
1,\dots,N\right\}\). The risk
$$
R(\alpha) = \sum_{h,d} L\left(h, \alpha\left(d\right)\right) P(h, d)
$$
can then be approximated by the empirical
risk,
$$
R_{emp}(\alpha) = \frac{1}{N} \sum_{i=1}^{N} L\left(h_i, \alpha\left(d_i\right)\right)
$$
Some methods used in machine learning, such as certain "neural networks" and support
vector machines, attempt to learn the decision rule directly by minimizing
\(R_{emp}(\alpha)\) instead of trying to model \(P(h, d)\).
More importantly for us, BRAINS may have evolved to apply either or
both of these two approaches in the context of a particular class of
tasks. The distinction between them is similar to the one between
"model-based" and "model-free" reinforcement learning, which I'll
discuss in Lecture 7.2.
EXTRA: Bayes Theorem helps unify classification and regression (Bishop, 2006, pp.196-199)
EXTRA: Here's how a classification problem can be reformulated as a regression
problem.
Consider the case of two classes, \({\cal C}_1\) and \({\cal C}_2\). The
posterior probability for class \({\cal C}_1\) can be written as
$$
\begin{align}
p({\cal C}_1\mid \textbf{x}) &= \frac{p(\textbf{x}\mid {\cal C}_1) p({\cal
C}_1)}{p(\textbf{x}\mid {\cal C}_1)p({\cal C}_1) + p(\textbf{x}\mid {\cal
C}_2)p({\cal C}_2)} \\
&= \frac{1}{1+exp(-a)} = \sigma(a)
\end{align}
$$
where \(a\) is the log likelihood ratio
$$
a = \ln \frac{p(\textbf{x}\mid {\cal C}_1)p({\cal C}_1)}{p(\textbf{x}\mid
{\cal C}_2)p({\cal C}_2)}
$$
and \(\sigma(a)\) is the logistic sigmoid function, defined by
$$
\sigma(a) = \frac{1}{1+exp(-a)}
$$
[EXTRA] an immediate application of the Bayes Theorem (cont.)
Now let's assume that the class-conditional densities are
\(D\)-dimensional Gaussian with the same covariance matrix \(\Sigma\):
$$
p(\textbf{x}\mid {\cal C}_k) = \frac{1}{(2\pi)^{D/2}}
\frac{1}{|\Sigma|^{1/2}} exp\left\{-\frac{1}{2} (\textbf{x}
-\mu_k)^{T} \Sigma^{-1} (\textbf{x} -\mu_k)\right\}
$$
For two classes, \(k=2\), the expression for \(p({\cal C}_1\mid
\textbf{x})\) from the previous slide yields:
$$
p({\cal C}_1\mid \textbf{x}) = \sigma(\textbf{w}^{T}\textbf{x} + w_0)
$$
where
$$
\textbf{w} = \Sigma^{-1}(\mu_1 - \mu_2)
$$
and
$$
w_0 = - \frac{1}{2} \mu_1^{T} \Sigma^{-1} \mu_1 +
\frac{1}{2} \mu_2^{T} \Sigma^{-1} \mu_2 +
\ln\frac{p({\cal C}_1)}{p({\cal C}_2)}
$$
[EXTRA] an immediate application of the Bayes Theorem (cont.)
CLASSIFICATION reformulated as REGRESSION:
$$
p({\cal C}_1\mid \textbf{x}) = \sigma(\textbf{w}^{T}\textbf{x} + w_0)
$$
The quadratic terms in \(\textbf{x}\) from the exponents of
the Gaussian densities have cancelled (due to the assumption of common
covariance matrices) leading to a linear function of \(\textbf{x}\) in the
argument of the logistic sigmoid.
Top: the class-conditional densities for two classes,
red and blue.
Bottom: the corresponding posterior probability
\(p({\cal C}_1\mid x)\), given by a logistic sigmoid of a linear
function of \(\textbf{x}\).
The surface on the right is colored
using a proportion of red ink given by \(p({\cal C}_1\mid \textbf{x})\)
and a proportion of blue ink given by \(p({\cal C}_2\mid \textbf{x}) = 1 -
p({\cal C}_1\mid \textbf{x})\).
