Intelligent Systems

Intelligent Systems

Decision Making:

1. Bayesian Decision Theory

2. Non-Bayesian Formulations

Recognition (recap)

The model

Let two random variables be given:

• The first one (𝑘 ∈ 𝐾) is typically discrete and is called “class”

• The second one (𝑥 ∈ 𝑋) is general (often continuous) and is called “observation”

Let the joint probability distribution 𝑝(𝑥, 𝑘) be “given”.

As 𝑘 is discrete it is often specified by 𝑝 𝑥, 𝑘 = 𝑝 𝑘 ⋅ 𝑝(𝑥|𝑘) The recognition task: given 𝑥, estimate 𝑘.

Usual problems (questions):

• How to estimate 𝑘 from 𝑥 (today)?

• The joint probability is not always explicitly specified.

• The set 𝐾 is sometimes huge.

Idea – a game

Somebody samples a pair according to a p.d.

He keeps hidden and presents to you

You decide for some according to a chosen decision strategy

Somebody penalizes your decision according to a Loss-function, i.e.

he compares your decision to the “true” hidden

You know both and the Loss-function (how does he compare) Your goal is to design the decision strategy in order to pay as less as possible in average.

Bayesian Risk

Notations:

The decision set . Note: it needs not to coincide with !!!

Examples – decisions “I don’t know”, “surely not this class” etc.

Decision strategy (mapping) Loss-function

The Bayesian Risk – the expected loss:

(should be minimized with respect to the decision strategy)

Some special cases

General:

Almost always:

decisions can be made for different independently (the set of decision strategies is not restricted). Then:

Very often:

the decision set coincides with the set of classes, i.e.

Maximum A-posteriori Decision (MAP)

The Loss is the simplest one:

i.e. we pay 1 if the answer is not the true class, no matter what error we make.

From that follows:

A MAP example

Let be given. Conditional probability distributions for observations given classes are Gaussians:

The loss-function is , i.e. we want MAP.

The decision strategy (the mapping ) partitions the input space into two regions: the one corresponding to the first and the one corresponding to the second class. How this partition looks like?

A MAP example

For a particular we decide for 1, if

Special case (for simplicity)

→ the decision strategy is (derivation on the board)

→ a linear classifier – the hyperplane that is orthogonal to More classes, equal and → Voronoi-diagram

More classes, equal and different → Fischer-classifier Two classes, different → a quadratic curve

etc.

Decision with rejection

The decision set is , i.e. extended by a special decision

“I don’t know”. The loss-function is

− we pay a (reasonable) penalty if we are lazy to decide.

Case-by-case analysis:

1. We decide for a class , decision is MAP , the loss for this is

2. We decide to reject , the loss for this is

→ Compare with and decide for the variant with

Other simple loss-functions

Let the set of classes be structured Example:

The probability distribution is with observations and

continuous hidden value . Suppose, we know for a given for which we would like to infer .

The Bayesian Risk reads:

Other simple loss-functions

Simple delta-loss-function → MAP (not interesting anymore) Loss may account for differences between the decision and the

“true” hidden value, for instance i.e. we pay depending on the distance. Than (see board again):

Other choices:

, combination with

Non-Bayesian decision making

Despite the generality of Bayesian approach, there are many tasks which cannot be expressed within the Bayesian framework:

• It is difficult to establish a penalty function, e.g. it does not assume values from the totally ordered set.

• A priori probabilities 𝑝 𝑘 are not known or cannot be known because 𝑘 is not a random event.

An example – Russian fairy tales hero

When he turns to the left, he loses his horse, when he turns to the right, he loses his sword, and if he turns back, he loses his beloved girl.

Is the sum of 𝑝₁ horses and 𝑝₂ swords is less or more than 𝑝₃ beloved girls?

Example: decisions while curing a patient

We have:

𝑥 ∈ 𝑋 – observations (features) measured on a patient 𝑘 ∈ 𝐾 = {ℎ𝑒𝑎𝑙𝑡ℎ𝑦, 𝑠𝑒𝑟𝑖𝑜𝑢𝑠𝑙𝑦 𝑠𝑖𝑐𝑘} – hidden states 𝑑 ∈ 𝐷 = {𝑑𝑜 𝑛𝑜𝑡 𝑐𝑢𝑟𝑒, 𝑎𝑝𝑝𝑙𝑦 𝑎 𝑑𝑟𝑢𝑔} – decisions Penalty function 𝐶: 𝐾 × 𝐷 → ?

Penalty problem – how to assign real number to a penalty?

𝐾\D 𝑑𝑜 𝑛𝑜𝑡 𝑐𝑢𝑟𝑒 𝑎𝑝𝑝𝑙𝑦 𝑎 𝑑𝑟𝑢𝑔

ℎ𝑒𝑎𝑙𝑡ℎ𝑦 Correct decision small health damage 𝑠𝑒𝑟𝑖𝑜𝑢𝑠𝑙𝑦 𝑠𝑖𝑐𝑘 death possible Correct decision

An example – enemy or allied airplane?

Observation 𝑥 describes the observed airplane.

Two hidden states 𝑘 = 1 allied airplane 𝑘 = 2 enemy airplane

The conditional probability 𝑝(𝑥|𝑘) can depend on the observation 𝑥 in a complicated manner but it exists and describes dependence of the observation 𝑥 on the situation 𝑘 correctly.

A priori probabilities 𝑝(𝑘) are not known and even cannot be known in principle.

→ the hidden state 𝑘 is not a random event.

Neyman-Pearson task

Observation 𝑥 ∈ 𝑋, two states: 𝑘 = 1 normal

𝑘 = 2 𝑑𝑎𝑛𝑔𝑒𝑟𝑜𝑢𝑠

The probability distribution of the observation 𝑥 depends on the state 𝑘 to which the object belongs, 𝑝 𝑥 𝑘 are known.

Given observation 𝑥, the task is to decide if the object is in the normal or dangerous state.

The set 𝑋 is to be partitioned into two such subsets 𝑋₁ (normal states) and 𝑋₂ (dangerous states), 𝑋 = 𝑋₁ ∪ 𝑋₂, 𝑋₁ ∩ X₂ = ∅ Note: the observation 𝑥 can belong to both states → there is no faultless strategy.

Neyman-Pearson task

The strategy is characterized by two numbers:

1. “Probability” of the false positive (false alarm) 𝜔 1 =

𝑥∈𝑋₂

𝑝(𝑥|1)

2. “Probability” of the false negative (overlooked danger) 𝜔 2 =

𝑥∈𝑋₁

𝑝(𝑥|2)

→ minimize the conditional probability of the false positive subject to the condition that the false negative is bounded:

𝑥∈𝑋₂

𝑝(𝑥|1) → min

𝑋₁,𝑋₂

𝑠. 𝑡.

𝑥∈𝑋₁

𝑝(𝑥|2) ≤ 𝜀

Neyman-Pearson task

Solution: Neyman–Pearson (1928, 1933)

The optimal strategy separates observation sets 𝑋₁ and 𝑋₂ according to a likelihood ratio by a threshold value 𝜃:

𝑄^∗ =

𝑘 = 1 𝑖𝑓 𝒑(𝒙|𝟏)

𝒑(𝒙|𝟐) > 𝜃 𝑘 = 2 otherwise

Other interesting non-Bayesian formulations:

1. Generalised Neyman-Pearson task for two dangerous states 2. Minimax task

3. Wald task 4. Linnik tasks

Conclusion and Outlook

Before:

1. Probability Theory 2. Decision making:

1. Bayesian Decision Theory: loss, risk …

2. Non-Bayesian formulations: Neyman-Pearson task … Next topics:

1. Probabilistic and discriminative learning (till 15.01) 2. Undirected graphical models (after 22.01)

Merry Christmas and happy New Year !!!