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Lecture 5: Representing uncertain knowledge
Prof. Gilles Louppe
g.louppe@uliege.be
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R: https://kuleshov.github.io/cs228-notes/representation/directed/ R: add a slide on how to "read" a bayesian network (and how to infer some independence properties) R: add d-separation https://inst.eecs.berkeley.edu/~cs188/fa18/assets/slides/lec14/FA18_cs188_lecture14_bayes_nets_II_independence_1pp.pdf
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- Probability:
- Random variables
- Joint and marginal distributions
- Conditional distributions
- Product rule, Chain rule, Bayes' rule
- Inference
- Bayesian networks:
- Representing uncertain knowledge
- Semantics
- Construction
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Do not overlook this lecture!
You will need these concepts a lot. (Now and in future courses)
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.grid[ .kol-1-2[ A ghost is hidden in the grid somewhere. Sensor readings tell how close a square is to the ghost.
- On the ghost: red
- 1 or 2 away: orange
- 3 away: yellow
- 4+ away green
Sensors are noisy, but we know the probability 
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[Q] Could we use a logical agent for this game?
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General situation:
- Observed variables (evidence): agent knows certain things about the state of the world (e.g., sensor readings).
- Unobserved variables: agent needs to reason about other aspects that are uncertain (e.g., where the ghost is).
- (Probabilistic) model: agent knows or believes something about how the known variables relate to the unknown variables.
How to handle uncertainty?
- A purely logical approach either:
- risks falsehood (because of ignorance about the world or laziness in the model), or
- leads to conclusions that are too weak for decision making.
- Probabilistic reasoning provides a framework for managing our knowledge and beliefs.
- Probabilistic assertions express the agent's inability to reach a definite decision regarding the truth of a proposition.
- Probability values summarize effects of
- laziness (failure to enumerate all world states)
- ignorance (lack of relevant facts, initial conditions, correct model, etc).
- (Bayesian subjective) Probabilities relate propositions to one's own state of knowledge (or lack thereof).
- e.g.,
$P(\text{ghost in cell } [3,2]) = 0.02$
- e.g.,
- These are not claims of a "probabilistic tendency" in the current situation (but might be learned from past experience of similar situations).
Begin with a set
A probability space is a sample space equipped with an assignment
- 1st axiom:
$P(\omega) \in \mathbb{R}$ ,$0 \leq P(\omega)$ for all$\omega \in \Omega$ . - 2nd axiom:
$P(\Omega) = 1$ . - 3rd axiom:
$P(\{ \omega_1, ..., \omega_n \}) = \sum_{i=1}^n P(\omega_i)$ for any set of samples.
where
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$\Omega$ = the 6 possible rolls of a die. -
$\omega_i$ (for$i=1, ..., 6$ ) are the sample points, each corresponding than an outcome of the die. - Assignment
$P$ for a fair die:$$P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \frac{1}{6}$$
- A random variable is a function
$X: \Omega \to D_X$ from the sample space to some domain defining its outcomes.- e.g.,
$Odd: \Omega \to \{ \text{true}, \text{false} \}$ such that$Odd(\omega) = (\omega % 2 = 1)$ .
- e.g.,
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$P$ induces a probability distribution for any random variable$X$ .$P(X=x_i) = \sum_{\{\omega: X(\omega)=x_i\}} P(\omega)$ - e.g.,
$P(Odd=\text{true}) = P(1)+P(3)+P(5) = \frac{1}{2}$ .
- In practice, we will use random variables to represent aspects of the world about which we (may) have uncertainty.
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$R$ : Is it raining? -
$T$ : Is it hot or cold? -
$L$ : Where is the ghost?
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For discrete variables, the probability distribution can be encoded by a discrete list of the probabilities of the outcomes, known as the probability mass function.
One can think of the probability distribution as a table that associates a probability value to each outcome of the variable.
- This table can be infinite!
- By construction, probability values are normalized (i.e., sum to
$1$ ).
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- Random variables are written in upper roman letters:
$X$ ,$Y$ , etc. - Realizations of a random variable are written in corresponding lower case letters.
E.g.,
$x_1$ ,$x_2$ , ...,$x_n$ could be of outcomes of the random variable$X$ . - The probability value of the realization
$x$ is written as$P(X=x)$ . - When clear from context, this will be abbreviated as
$P(x)$ . - The probability distribution of the random variable
$X$ is denoted as$P(X)$ . This corresponds e.g. to a full probability table (and not to a single scalar value!).
A joint probability distribution over a set of random variables
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An event is a set
$P(E) = \sum_{(x_1, ..., x_n) \in E} P(X_1=x_1, ..., X_n=x_n)$
From a joint distribution, the probability of any event can be calculated.
- Probability that it is hot and sunny?
- Probability that it is hot?
- Probability that it is hot or sunny?
Interesting events often correspond to partial assignments.
- e.g.,
$P(T=\text{hot})$
The marginal distribution of a subset of a collection of random variables is the joint probability distribution of the variables contained in the subset.
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Intuitively, marginal distributions are sub-tables which eliminate variables.
[Q] To what events are marginal probabilities associated?
Prior or unconditional probabilities of an event correspond to some initial belief, prior to arrival of any evidence.
- e.g.,
$P(W=\text{sun}) = 0.6$ .
Posterior or conditional probabilities correspond to the probability of an event, given some observed evidence.
Formally,
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Draw the Venn diagram of p(a), p(b), p(a,b) and explain p(a|b). ^ add that plot
(A and B are boolean rvs defined on Omega)
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Conditional distributions are probability distributions over some variables, given fixed values for others.
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[Q] To what events are conditional probabilities associated?
[Q] Is a conditional distribution defined on the same sample and probability space than the joint?
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Select the joint probabilities matching the evidence
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Normalize the selection (make it sum to
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[Q] Why does this work?
Probabilistic inference is the problem of computing a desired probability from other known probabilities (e.g., conditional from joint).
- We generally compute conditional probabilities.
- e.g.,
$P(\text{on time} | \text{no reported accidents}) = 0.9$ - These represent the agent's beliefs given the evidence.
- e.g.,
- Probabilities change with new evidence:
- e.g.,
$P(\text{on time} | \text{no reported accidents}, \text{5AM}) = 0.95$ - e.g.,
$P(\text{on time} | \text{no reported accidents}, \text{rain}) = 0.8$ - e.g.,
$P(\text{ghost in } [3,2] | \text{red in } [3,2]) = 0.99$ - Observing new evidence causes beliefs to be updated.
- e.g.,
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Evidence variables:
$E_1, ..., E_k = e_1, ..., e_k$ -
Query variables:
$Q$ -
Hidden variables:
$H_1, ..., H_r$ -
$(Q \cup E_1, ..., E_k \cup H_1, ..., H_r)$ = all variables$X_1, ..., X_n$
Inference is the problem of computing
Start from the joint distribution
- Select the entries consistent with the evidence
$E_1, ..., E_k = e_1, ..., e_k$ . - Marginalize out the hidden variables to obtain the joint of the query and the evidence variables:
$$P(Q,e_1,...,e_k) = \sum_{h_1, ..., h_r} P(Q, h_1, ..., h_r, e_1, ..., e_k).$$ - Normalize:
$$\begin{aligned} Z &= \sum_q P(q,e_1,...,e_k) \\\\ P(Q|e_1, ..., e_k) &= \frac{1}{Z} P(Q,e_1,...,e_k) \end{aligned}$$
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$P(W)$ ? -
$P(W|\text{winter})$ ? -
$P(W|\text{winter},\text{hot})$ ?
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- Inference by enumeration can be used to answer probabilistic queries for discrete variables (i.e., with a finite number of values).
- However, enumeration does not scale!
- Assume a domain described by
$n$ variables taking at most$d$ values. - Space complexity:
$O(d^n)$ - Time complexity:
$O(d^n)$
- Assume a domain described by
- Can we reduce the size of the representation of the joint distribution?
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More generally, any joint distribution can always be written as an incremental product of conditional distributions:
[Q] Why is this always true?
-
$P(a|b) = P(a)$ , or -
$P(b|a) = P(b)$ , or $P(a,b) = P(a)P(b)$
Independence is also written as
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The original 32-entry table reduces to one 8-entry and one 4-entry table (assuming 4 values for
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For
$2^n \to n$
[Q] In practice, absolute independence is rare. What can we do?
-
$P(a|b,c) = P(a|c)$ , or -
$P(b|a,c) = P(b|c)$ , or $P(a,b|c) = P(a|c)P(b|c)$
Conditional independence is also written as
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- Using the chain rule, the join distribution can be factored as a product of conditional distributions.
- Each conditional distribution may potentially be simplified by conditional independence.
- Conditional independence assertions allow probabilistic models to scale up.
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- Assume three random variables
$\text{Toothache}$ ,$\text{Catch}$ and$\text{Cavity}$ . -
$\text{Catch}$ is conditionally independent of$\text{Toothache}$ , given$\text{Cavity}$ .
Therefore, we can write:
In this case, the representation of the joint distribution reduces to
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- More generally, from the product rule, we have:
$P(\text{cause},\text{effect}_1, ..., \text{effect}_n) = P(\text{effect}_1, ..., \text{effect}_n|\text{cause}) P(\text{cause})$
- Assuming pairwise conditional independence between the effects given the cause, it comes:
$P(\text{cause},\text{effect}_1, ..., \text{effect}_n) = P(\text{cause}) \prod_i P(\text{effect}_i|\text{cause}) $
- This probabilistic model is called a naive Bayes model.
- The complexity of this model is
$O(n)$ instead of$O(2^n)$ without the conditional independence assumptions. - Naive Bayes can work surprisingly well in practice, even when the assumptions are wrong.
- The complexity of this model is
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The next slide is .bold[super] important.
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The product rule defines two ways to factor the joint distribution of two random variables.
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$P(a)$ is the prior belief on$a$ . -
$P(b)$ is the probability of the evidence$b$ . -
$P(a|b)$ is the posterior belief on$a$ , given the evidence$b$ . -
$P(b|a)$ is the conditional probability of$b$ given$a$ . Depending on the context, this term is called the likelihood.
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Why is this helpful?
- The Bayes rule let us build one conditional from its reverse.
- Often one conditional is tricky, but the other is simple.
This equation is the foundation of many AI systems.
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$P(\text{effect}|\text{cause})$ quantifies the relationship in the causal direction. -
$P(\text{cause}|\text{effect})$ describes the diagnostic direction.
Let
$P(s|m) = 0.7$ $P(m) = 1/50000$ $P(s) = 0.01,$
it comes
$P(m|s) = \frac{P(s|m)P(m)}{P(s)} = \frac{0.7 \times 1/50000}{0.01} = 0.0014.$
- Let assume a random variable
$G$ for the ghost location and a set of random variables$R_{i,j}$ for the individual readings. - We define a prior distribution
$P(G)$ over ghost locations.- Assume it is uniform.
- We assume a sensor reading model
$P(R_{i,j} |G)$ .- That is, we know what the sensors do.
-
$R_{i,j}$ = reading color measured at$[i,j]$ - e.g.,
$P(R_{1,1}=\text{yellow}|G=[1,1])=0.1$
- e.g.,
- Two readings are conditionally independent, given the ghost position.
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- We can calculate the posterior distribution
$P(G|R_{i,j})$ using Bayes' rule:$$P(G|R_{i,j}) = \frac{P(R_{i,j}|G)P(G)}{P(R_{i,j})}.$$ - For the next reading
$R_{i',j'}$ , this posterior distribution becomes the prior distribution over ghost locations, which we update similarly.
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[Q] What if we had chosen a different prior?
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What do probability values represent?
- The objectivist frequentist view is that probabilities are real aspects of the universe.
- i.e., propensities of objects to behave in certain ways.
- e.g., the fact that a fair coin comes up heads with probability
$0.5$ is a propensity of the coin itself.
- The subjectivist Bayesian view is that probabilities are a way of characterizing an agent's beliefs or uncertainty.
- i.e., probabilities do not have external physical significance.
- This is the interpretation of probabilities that we will use!
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- The joint probability distribution can answer any question about the domain.
- However, its representation can become intractably large as the number of variable grows.
- Independence and conditional independence reduce the number of probabilities that need to be specified in order to define the full joint distribution.
- These relationships can be represented explicitly in the form of a Bayesian network.
A Bayesian network is a directed graph in which:
- Each node corresponds to a random variable.
- Can be observed or unobserved.
- Can be discrete or continuous.
- Each edge indicates "direct influence" between variables.
- If there is an arrow from node
$X$ to node$Y$ ,$X$ is said to be a parent of$Y$ . - The graph has no directed cycle (i.e., the graph is a DAG).
- If there is an arrow from node
- Each node
$X_i$ is annotated with a conditional probability distribution$P(X_i | \text{parents}(X_i))$ that quantifies the effect of the parents on the node.- In the simplest case, conditional distributions are represented as conditional probability table (CTP).
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The topology of the network encodes conditional independence assertions:
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$\text{Weather}$ is independent of the other variables. -
$\text{Toothache}$ and$\text{Catch}$ are conditionally independent given$\text{Cavity}$ .
[Q] What is the BN for
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In a Naive Bayes model, we assume pairwise conditional independence between the effects given the cause:
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I am at work, neighbor John calls to say my alarm is ringing, but neighbor Mary does not call. Sometimes it's set off by minor earthquakes. Is there a burglar?
- Variables:
$\text{Burglar}$ ,$\text{Earthquake}$ ,$\text{Alarm}$ ,$\text{JohnCalls}$ ,$\text{MaryCalls}$ . - Network topology from "causal" knowledge:
- A burglar can set the alarm off
- An earthquake can set the alaram off
- The alarm can cause Mary to call
- The alarm can cause John to call
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A Bayesian network implicitly encodes the full joint distribution as the product of the local distributions:
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Why does
- By the chain rule,
$P(x_1, ..., x_n) = \prod_{i=1}^n P(x_i | x_1, ..., x_{i-1})$ . - Assume conditional independencies of
$X_i$ with its predecessors in the ordering given the parents, and provided$\text{parents}(X_i) \subseteq \{ X_1, ..., X_{i-1}\}$ :$$P(x_i | x_1, ..., x_{i-1}) = P(x_i | \text{parents}(X_i))$$ - Therefore
$P(x_1, ..., x_n) = \prod_{i=1}^n P(x_i | \text{parents}(X_i))$ .
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A node
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A node
Bayesian networks are correct representations of the domain only if each node is conditionally independent of its other predecessors in the node ordering, given its parents.
This suggests the following methodology for building Bayesian networks:
- Choose an ordering of variables
$X_1, ..., X_n$ . - For
$i=1$ to$n$ :- Add
$X_i$ to the network. - Select a minimal set of parents from
$X_1, ..., X_{i-1}$ such that$P(x_i | x_1, ..., x_{i-1}) = P(x_i | \text{parents}(X_i))$ . - For each parent, insert a link from the parent to
$X_i$ . - Write down the CPT.
- Add
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These networks are alternatives to Example 3. ]
[Q] What do you think of these BNs?
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For the left network:
- P (J|M ) = P (J)? No
- P (A|J, M ) = P (A|J)? P (A|J, M ) = P (A)? No
- P (B|A, J, M ) = P (B|A)? Yes
- P (B|A, J, M ) = P (B)? No
- P (E|B, A, J, M ) = P (E|A)? No
- P (E|B, A, J, M ) = P (E|A, B)? Yes
- A CPT for boolean
$X_i$ with$k$ boolean parents has$2^k$ rows for the combinations of parent values. - Each row requires one number
$p$ for$X_i = true$ .- The number of
$X_i=false$ is just$1-p$ .
- The number of
- If each variable has no more than
$k$ parents, the complete network requires$O(n \times 2^k)$ numbers.- i.e., grows linearly with
$n$ , vs.$O(2^n)$ for the full joint distribution.
- i.e., grows linearly with
- For the burglary net, we need
$1+1+4+2+2=10$ numbers (vs.$2^5-1=31$ ). - Compactness depends on the node ordering.
- Compare the three networks representing the same joint distribution.
- When the network reflects the true causal patterns:
- Often more compact (nodes have fewer parents).
- Often easier to think about.
- Often easier to elicit from experts.
- But, Bayesian networks need not be causal.
- Sometimes no causal network exists over the domain (e.g., if variables are missing).
- Edges reflect correlation, not causation.
- What do the edges really mean then?
- Topology may happen to encode causal structure.
- Topology really encodes conditional independence.
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Correlation does not imply causation.
Causes cannot be expressed in the language of probability theory.
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Philosophers have tried to define causation in terms of probability:
However, the inequality
- Instead, the expression means that if we observe
$X$ , then the probability of$Y$ increases. - But this increase may come about for other reasons!
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The correct formulation should read
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- The reading in barometer is useful to predict rain.
$$P(\text{rain}|\text{Barometer}=\text{high}) > P(\text{rain}|\text{Barometer}=\text{low})$$ - But hacking a barometer will not cause rain!
$$P(\text{rain}|\text{Barometer hacked to high}) = P(\text{rain}|\text{Barometer hacked to low})$$
- Uncertainty arises because of laziness and ignorance. It is inescapable in complex non-deterministic or partially observable environments.
- Probabilistic reasoning provides a framework for managing our knowledge and beliefs.
- Bayesian networks are DAGs whose nodes correspond to random variables; each node has a conditional distribution for the node, given its parents.
- A Bayesian Network specifies a full joint distribution.
- They are often exponentially smaller than an explicitly enumerated joint distribution.
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