Beliefs and credences

class: center, middle, inverse, title-slide

.title[
# Beliefs and credences
]
.subtitle[
## How To Think - Week 4
]
.author[
### Fernando Alvear
]
.institute[
### University of Missouri
]
.date[
### Feb 6
]

---

# Accuracy

One goal of good reasoning is _accuracy_:

- Our views should be _accurate_: they should reflect how things really are.

---
class: center, middle

Have you ever _changed your mind_ about something? How was that process?

---
class: center, middle

Do you ever regret _not having changed your mind_ about something? What could you have done instead to make that happen?

---
# Mindset

Reasoning well is about cultivating a _mindset._
 
| Attitude | Elements |
|----------|-----------|
| Be curious    | <ul><li>Adopt the mentality of the _scout_ instead of the _soldier_ in reasoning. ✅</li></ul>|
| Be thorough | <ul><li>Look for alternative answers/explanations. ✅</li><li>Look for additional evidence. ✅</li></ul>|
| Be open | <ul><li>Consider the opposite. ✅</li><li>Revise your degrees of confidence.</li></ul>|

---
class: middle, center

# Revise your _degrees of confidence_.

What does this mean?

---
# Some terminology

**Proposition**: A complete sentence* expressing a circumstance or state of affairs. They are sometimes called _statements_. They are the primary bearers of truth values (truth and falsity).

Some propositions:
- Today is Monday.
- Nico has seven toys.
- 2+2=5.
- Unicorns poop rainbow-colored ice cream.

Some sentences that are not propositions:
- Shut the door!
- Is the window closed?
- Nico and his toys
- 2+2

(*): Technically, propositions are not sentences. They are _what sentences are about (or refer to)_. But we will ignore this wrinkle (and many others) for the moment.

---
class: center, middle
background-image: url('assets/squatty-potty.jpg')

---
# Beliefs and degrees of confidence

**Belief**: A propositional attitude we have when we take something to be the case or regard it as true.

> I believe that the trash will be picked up today.

> I believe that 2+2=4

**Credence**: A propositional attitude we have whenever we have a _partial_ confidence that something is the case. They are also called "degrees of confidence" and can be expressed as probabilities (a real number from 0 to 1).

> My credence that the trash will be picked up today is 0.95

> My credence that 2+2=4 is 1

.center[
<img src="assets/degrees-of-belief.jpeg" alt="" height="170"/>]
---

# Belief-like states

**Belief**: A propositional attitude we have when we take something to be the case or regard it as true.

**Disbelief**: A propositional attitude we have when we take something to _not_ be the case or regard it as _false_.

**Suspension of judgment**: A propositional attitude we have when we remain agnostic or undecided regarding a proposition (we neither believe nor disbelieve it).

Examples:

- I believe that God exists.

- I disbelieve that the Earth is flat.

- I believe that that the Earth is _not_ flat.

- I suspend judgment regarding the proposition that tomorrow will be sunny.

---

# Credences

Examples:

- My credence that I will roll a 4 with a single die is 1/6.
- My credence that the toss of a fair coin will land heads is 1/2 (or .5)
- My credence that God exists is 2/3
- My credence that 2+2=4 is 1.
- My credence that 2+2=5 is 0.

There are _infinite_ credence-states: From 0 to 1. (or 0% to 100%)

.center[
<img src="assets/probability-line.svg" alt="" height="170"/>]

---

# Credences

True propositions should receive credence 1.

False propositions should receive credence 0.

- My credence that 2+2=4 is 1.
- My credence that 2+2=5 is 0.

But this can occur only when we attain _certainty_. In conditions of _uncertainty_, credences reflect the support we have in favor of the proposition:

- My credence that I will roll a 4 with a single die is 1/6.
- My credence that my trash will be picked up today is 99%
- My credence that the toss of a fair coin will land heads is 1/2 (or .5)
- My credence that God exists is 2/3

.center[
<img src="assets/probability-line.svg" alt="" height="170"/>]

---

# Credences and probability

Credences use the language of _probability._

$$ \text{Probability of an event happening} = \frac{\text{Number of ways it can happen}}{\text{Total number of outcomes}} $$
Six-sided die .font-big[🎲]
- Six possible outcomes: .font-big[⚀ ⚁ ⚂ ⚃ ⚄ ⚅]
- One way that rolling a 4 can happen: .font-big[⚃]
$$ P(\text{rolling a 4}) = \frac{1}{6} $$

Before rolling the die, my degree of confidence that I will roll a 4 should be 1/6.

---
# Expressing probabilities

Decimals:
.center[
<img src="assets/probability-line-decimal.svg" alt="" width="500"/>]
Fractions:
.center[
<img src="assets/probability-line-fracs.svg" alt="" width="500"/>]
Percents:
.center[
<img src="assets/probability-line-percent.svg" alt="" width="500"/>]

---
# Expressing probabilities

- Chances

- "The chances of rolling a 4 are 1 in 6"
  
- Odds

- "The odds of rolling a 4 are 1 to 5"
  
  - "One person has disease X for every thousand people who don’t..."
  
- Combination of chances and odds

- There's a _50-50 chance_ that the coin will land heads.

---

# Imprecise probabilities in ordinary language

- It's _likely / probable_ that tomorrow will be sunny.

- There's a _good / very good chance_ that you will pass this class.

- Greater than 50%

- It's _unlikely / improbable_ that tomorrow will snow.

- There's a _remote / low chance_ that you will fail this class.

- Lower than 50%

- It's a _toss-up_ between Angela and Mary for the editor's job.

- It's a _toss-up_ between Angela and Moira for the editor's job.

- Around 50%

---
# Beliefs, credences, and accuracy

Our goal is to have _accurate_ beliefs and degrees of confidence.

- Accurate beliefs:

- A belief is accurate if it's true.
  - A belief is inaccurate if it's false.

- Accurate degrees of confidence:

- The shorter the distance between our confidence and the truth-value of the proposition (either 0 or 1), the more accurate it is.

---
class: middle, center

# Why think in terms of credences?

---
# Why think in terms of credences?

Our goal is to have _accurate_ beliefs and degrees of confidence.

Perhaps we should use S2 (and not S1) to form our beliefs and degrees of confidence, so they follow the evidence.

But do we have _direct control_ over our beliefs? Can someone believe _at will_?

For those who think we do have direct control over our beliefs... please believe the following:

T: A ferocious tiger is inside this classroom.

Do you believe in T? If so, why you are not running for your life?

---
# Indirect control over beliefs

- We **don't** have _direct_ control over our beliefs.

- Sometimes we have overwhelming evidence that something is true and we can't avoid believing in it. This is a _good_ reasoning process coming from S1.

- Example: Belief that the lights are on.

- Other times, we fall in cognitive pitfalls. This is a _bad_ reasoning process coming from S1.

- Example: Heuristics, confirmation bias, visual illusions.

- Yet other times, especially when evidence is not 100% clear, we can _indirectly guide our beliefs_ by reflecting and reasoning. This is S2 acting, but it can go either way.

- Example: Tobacco causes cancer. ✅
  - Example: 5G causes cancer. ❌

---
# Why think in terms of credences?

- We cannot directly change our beliefs, but we can reflect about and change our degrees of confidence according to the evidence.

- Evidence → degree of confidence

- Our degrees of confidence can then influence our beliefs.

- Higher degree of confidence → belief
  
  - Middle-of-the scale degree of confidence → suspension

- Lower degree of confidence → disbelief

- Thinking about our degrees of confidence gives us a reliable way to influence our beliefs for the better.

- Evidence → degrees of confidence → belief/disbelief/suspension

---
class: middle, center

Consider proposition P: <br>"Objects belonging to intelligent extraterrestrial beings have visited Earth"

Do you believe that P?

What is your degree of confidence on P?

---
class: middle, center

---
class: middle, center

---
class: middle, center

Regarding proposition P: <br>"Objects belonging to intelligent extraterrestrial beings have visited Earth"

Upon seeing the video...

Do you believe that P _now_?

What is your _new_ degree of confidence on P?

---
# Revising our degrees of confidence

- The videos from the US navy provide _some_ evidence for P.

- If P were true, then it's very probable that the US Navy has footage like  the one we saw.
  
- But there's also evidence against P.

- If P were true, then we would have by now high-quality pictures or video of these objects. But we only have shaky video / low quality images.
  
  - Several other videos / images have been debunked.
  
  - Extraterrestrial travel seems very hard. If P were true, then our understanding of physics would be significantly incomplete.
  
- Our degree of confidence regarding P should increase upon seeing the video, but this change would probably not be enough to change our beliefs regarding P.

- Example: from 1% to 2%.

---
class: middle, center

# Why think in terms of credences?

Answer 1: They allow us to slightly update or revise our stances about the world according to the evidence.

Answer 2: They allow us to avoid believing with insufficient evidence.

---
# Revising our degrees of confidence

Proposition A: Aries people are mean, competitive, and rude.

John believes A, and assigns a high degree of confidence to it (let's say 90%).

John gets to know Sarah. She's nice and considerate. He learns that Sarah is Aries. Thus, John's credence in A changes to 60%.

John gets to know Matt. He's nice and considerate. He learns that Matt is Aries. Thus, John's credence in A changes to 30%

John reflects about how the relative position of the sun and the stars could determine a person's character. Since he doesn't find any plausible explanation for this, he concludes that this is very unlikely. Thus, John's credence in A changes to 1%.

John is asked "Do you believe in A?" John considers how his degree of confidence has diminished drastically since he started believing in A. Thus, John realizes he no longer believes in A.

---
# Revising our degrees of confidence

Proposition B: Skill and success is determined by innate talent, and not by practice and hard work.

Emma believes B, and assigns a high degree of confidence to it (let's say 90%).

Emma meets Julia. She's terrible at playing the violin. 10 years later, Emma recognizes Julia as the main violinist in an orchestra performance. She plays the violin beautifully. Thus, Emma's credence in B changes to 50%

Emma meets Susana and Taylor in high school. Susana doesn't study much and gets excellent grades. Taylor gets the same good grades but has to work a lot. Later, Emma encounters them 10 years later. Susana didn't finish college and works at a grocery store. Taylor finished college and is the director of an NGO. Thus, Emma's credence in B changes to 10%

Emma is asked "Do you believe in B?" Emma considers how her degree of confidence has diminished drastically since she started believing in B. Thus, Emma realizes she no longer believes in B.

---
# Revising our degrees of confidence

Emma and John, for whatever reason, started believing a false proposition.

Emma and John remained open to evidence and adjusted their degrees of confidence accordingly.

As evidence accumulates, Emma and John were able to update their degrees of confidence, which led them to adopt true (accurate) beliefs.

This suggests the following idea:

- **Convergence to truth**: As long as we remain open to receive evidence, evaluate it properly, and update our degrees of confidence accordingly, __most of our beliefs would end up being correct__ (in the long run).

---
class: middle, center

# Why think in terms of credences?

Answer 3: They allows us to attain accurate beliefs over time (through revising and openness).

Answer 4: They allow us to adopt beliefs only when we have sufficient evidence.

---
# Summary

- Two main propositional attitudes:
  - Beliefs
  - Degrees of confidence (or credences)
- Degrees of confidence use the language of probability.
- Our goal is to have accurate beliefs and accurate degrees of confidence.
- We should guide our degrees of confidence and beliefs according to the evidence.
- Since we can't _directly_ control our beliefs, we can use our degrees of confidence to _indirectly_ influence our beliefs.
- Degrees of confidence:
  - They allow us to slightly update or revise our stances about the world according to the evidence. 
  - They allow us to avoid believing with insufficient evidence.
  - They allow us to attain accurate beliefs over time (through revising and openness).
  - They allow us to adopt beliefs only when we have sufficient evidence.

---
# Mindset

Reasoning well is about cultivating a _mindset._

| Attitude | Elements |
|----------|-----------|
| Be curious    | <ul><li>Adopt the mentality of the _scout_ instead of the _soldier_ in reasoning. ✅</li></ul>|
| Be thorough | <ul><li>Look for alternative answers/explanations. ✅</li><li>Look for additional evidence. ✅</li></ul>|
| Be open | <ul><li>Consider the opposite. ✅</li><li>Revise your degrees of confidence. ✅</li></ul>|

---

# Question for this week's quiz

Have you ever _changed your mind_ about something? How was that process?  Describe an example of the process of changing one's view using the concepts of belief and of degrees of confidence.

---

# Questions for discussion

Does it make sense to believe a proposition but have a very low credence in it?

Does it make sense to disbelieve a proposition but have a very high credence in it?

What do you think about the following statements?

- I believe I will win the lottery tomorrow.
- I believe we will