Bayes basic

Bayes' Theorem is a fundamental concept in probability theory that describes how to update the probability of a hypothesis based on new evidence. It relates conditional probabilities and provides a mathematical framework for reasoning about uncertainty.

Formula:

The general formula for Bayes' Theorem is:
[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
]

Where:
- $P(A\mid B)$: The posterior probability of $A$, given $B$ (updated probability of $A$ after observing $B$).
- $P(B\mid A)$: The likelihood, or the probability of $B$ given that $A$ is true.
- $P(A)$: The prior probability of $A$ (initial belief about $A$).
- $P(B)$: The marginal probability of $B$, or the total probability of observing $B$.

Key Idea:

Bayes' Theorem allows you to update the probability of an event $A$ happening based on new evidence $B$.

---

Derivation:

Bayes' Theorem is derived from the definition of conditional probability:
[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
]
And:
[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
]
Rearranging $P(B\mid A)$, we get:
[
P(A \cap B) = P(B \mid A) \cdot P(A)
]
Substituting this into $P(A\mid B)$, we get:
[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
]

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Example:

Suppose a medical test for a disease is 99% accurate:
- If a person has the disease, the test is positive 99% of the time ($P(\text{Positive}\mid \text{Disease})=0.99$).
- If a person does not have the disease, the test is negative 95% of the time ($P(\text{Negative}\mid \text{No Disease})=0.95$).
- The disease occurs in 1% of the population ($P(\text{Disease})=0.01$).

You take the test, and it’s positive. What’s the probability you have the disease ($P(\text{Disease}\mid \text{Positive})$)?

Solution:

Using Bayes' Theorem:
[
P(\text{Disease} \mid \text{Positive}) = \frac{P(\text{Positive} \mid \text{Disease}) \cdot P(\text{Disease})}{P(\text{Positive})}
]

First, calculate $P(\text{Positive})$ using the law of total probability:
[
P(\text{Positive}) = P(\text{Positive} \mid \text{Disease}) \cdot P(\text{Disease}) + P(\text{Positive} \mid \text{No Disease}) \cdot P(\text{No Disease})
]
[
P(\text{Positive}) = (0.99 \cdot 0.01) + (0.05 \cdot 0.99) = 0.0099 + 0.0495 = 0.0594
]

Now, substitute into Bayes' Theorem:
[
P(\text{Disease} \mid \text{Positive}) = \frac{0.99 \cdot 0.01}{0.0594} = \frac{0.0099}{0.0594} \approx 0.1667
]

So, even with a positive test result, the probability of actually having the disease is about 16.67%, mainly because the disease is rare.

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can we say P(B|A) is posteriary

No, $P(B\mid A)$ is not referred to as the "posterior probability." Instead:

• $P(A\mid B)$ is called the posterior probability because it represents the updated probability of $A$ after observing $B$ (the evidence).
• $P(B\mid A)$ is called the likelihood, which represents the probability of observing $B$ given that $A$ is true.

To summarize in the context of Bayes' theorem:
[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
]
- $P(A)$: Prior probability (your initial belief about $A$).
- $P(B\mid A)$: Likelihood (how likely the evidence $B$ is if $A$ is true).
- $P(A\mid B)$: Posterior probability (updated belief about $A$ after observing $B$).
- $P(B)$: Marginal probability of $B$, normalizing factor for the probabilities.

Why $P(B\mid A)$ is not posterior

The posterior $P(A\mid B)$ directly answers the question: "Given the evidence $B$, what is the updated probability of $A$?" In contrast, $P(B\mid A)$ is a measure of how consistent the evidence $B$ is with the assumption that $A$ is true—it does not update or refine the belief about $A$.

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so with the help of priory and likelihood we update the posteriory

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Yes, exactly! Using prior and likelihood, we compute the posterior probability through Bayes' Theorem.

Key Components:

1. Prior ($P(A)$):
2. Represents your initial belief about the probability of event $A$ before observing any evidence.

3. Example: The probability of a person having a disease based on general population statistics.

4. Likelihood ($P(B\mid A)$):

5. Measures how consistent the evidence $B$ is with the hypothesis $A$.

6. Example: The probability of a positive test result if the person has the disease.

7. Posterior ($P(A\mid B)$):

8. The updated probability of $A$, given the observed evidence $B$.
9. It combines prior knowledge and the likelihood of the evidence.

10. Example: The updated probability of having the disease after getting a positive test result.

11. Marginal Probability ($P(B)$):

12. A normalizing constant ensuring the probabilities sum to 1.
13. It is the total probability of observing the evidence $B$, considering all possible causes.

Bayes' Theorem in Action:

[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
]
- Prior ($P(A)$): What you believe about $A$ before evidence $B$.
- Likelihood ($P(B\mid A)$): How well $A$ explains $B$.
- Together, they update the posterior ($P(A\mid B)$).

Intuitive Understanding:

• Prior: "Before seeing any data, this is my belief about $A$."
• Likelihood: "Given $A$, how likely is the evidence $B$?"
• Posterior: "After seeing $B$, this is my new belief about $A$."

Example:

Let’s revisit the medical test example:
- $P(\text{Disease})=0.01$ (Prior: 1% of the population has the disease).
- $P(\text{Positive}\mid \text{Disease})=0.99$ (Likelihood: If diseased, test is positive 99% of the time).
- Using Bayes' Theorem, we update the belief about having the disease given a positive result ($P(\text{Disease}\mid \text{Positive})$).

The posterior reflects our updated belief about the disease's probability after accounting for the test's reliability and initial assumptions.

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Bayes' Theorem fits naturally into the framework of generative models, as these models aim to describe how data is generated, often involving latent variables or hidden structures.

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Generative Models and Bayes' Theorem:

A generative model seeks to model the joint probability distribution $P(X,Z)$, where:
- $X$: Observed data (e.g., images, text, or sounds).
- $Z$: Latent variables or hidden factors (e.g., topics in text, noise in data).

Using Bayes' Theorem, the generative model can infer the hidden variables or causes ($Z$) from the observed data ($X$) by computing the posterior $P(Z\mid X)$.

---

How Bayes' Theorem Fits:

1. Generative Process:

2. A generative model defines $P(X,Z)=P(X\mid Z)P(Z)$, where:

   • $P(Z)$: Prior distribution over latent variables $Z$.
   • $P(X\mid Z)$: Likelihood or conditional distribution modeling how $Z$ generates $X$.

3. Inference with Bayes' Theorem:

4. To infer $Z$ given $X$ (i.e., to compute $P(Z\mid X)$), we apply Bayes' Theorem:
   [
   P(Z \mid X) = \frac{P(X \mid Z) \cdot P(Z)}{P(X)}
   ]

5. Here:

   • Prior ($P(Z)$): Encodes the model's assumptions about $Z$ before observing $X$.
   • Likelihood ($P(X\mid Z)$): Captures how the latent variable $Z$ generates the observed data $X$.
   • Posterior ($P(Z\mid X)$): Updates beliefs about $Z$ after observing $X$.

6. Marginal Likelihood ($P(X)$):

7. Normalizes the posterior and ensures it sums to 1.
8. Calculated as:
   [
   P(X) = \int P(X \mid Z) P(Z) \, dZ
   ]
9. In complex models, this integral can be computationally expensive and is often approximated.

---

Example in Generative Models:

1. Latent Dirichlet Allocation (LDA):

• In LDA (a generative model for topic modeling), the observed data $X$ are words in documents, and $Z$ are the latent topics.
• Bayes' Theorem is used to infer the posterior distribution of topics $P(Z\mid X)$, given observed words.

2. Variational Autoencoders (VAEs):

• A VAE is a deep generative model where:
  • $P(Z)$: Prior over latent variables.
  • $P(X\mid Z)$: Likelihood of reconstructing data $X$ given $Z$.
• The posterior $P(Z\mid X)$ is approximated using a variational distribution because direct computation is intractable.

3. GANs (Generative Adversarial Networks):

• While GANs are not explicitly Bayesian, they involve learning $P(X)$ indirectly by generating samples from an implicit $P(X\mid Z)$ and refining $P(Z)$.

---

Why Bayes' Theorem is Important for Generative Models:

1. Inference: Helps compute posteriors $P(Z\mid X)$, linking observed data to latent causes.
2. Uncertainty: Allows quantification of uncertainty in predictions or latent variables.
3. Learning: Bayesian principles underpin many generative learning methods, including variational inference and expectation-maximization.

In summary, Bayes' Theorem provides the theoretical foundation for making inferences in generative models, connecting data $X$ with latent variables $Z$ in a principled manner.

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Let's walk through numerical examples illustrating how Bayes' Theorem is applied in generative models. These examples demonstrate inferring latent variables ($Z$) from observed data ($X$).

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1. Topic Modeling in Latent Dirichlet Allocation (LDA)

Problem:

You have a document $X=\text{"apple banana orange"}$, and you're inferring the topic distribution $Z$. Suppose there are two possible topics:
- Topic 1: Fruits
- Topic 2: Technology

Generative Model Components:

• Prior ($P(Z)$):
  [
  P(Z = \text{Fruits}) = 0.6, \quad P(Z = \text{Technology}) = 0.4
  ]
• Likelihood ($P(X\mid Z)$):
  Word probabilities for each topic:
• If $Z=\text{Fruits}$: $P(\text{"apple banana orange"}\mid \text{Fruits})=0.5\times 0.4\times 0.6=0.12$
• If $Z=\text{Technology}$: $P(\text{"apple banana orange"}\mid \text{Technology})=0.1\times 0.05\times 0.02=0.0001$

Applying Bayes' Theorem:

We want to compute $P(Z\mid X)$ for both topics.

1. Compute $P(X)$ (normalizing factor):
   [
   P(X) = P(X \mid \text{Fruits}) \cdot P(\text{Fruits}) + P(X \mid \text{Technology}) \cdot P(\text{Technology})
   ]
   [
   P(X) = (0.12 \times 0.6) + (0.0001 \times 0.4) = 0.072 + 0.00004 = 0.07204
   ]

2. Compute posterior probabilities:

3. For $Z=\text{Fruits}$:
   [
   P(\text{Fruits} \mid X) = \frac{P(X \mid \text{Fruits}) \cdot P(\text{Fruits})}{P(X)} = \frac{0.12 \times 0.6}{0.07204} \approx 0.999
   ]
4. For $Z=\text{Technology}$:
   [
   P(\text{Technology} \mid X) = \frac{P(X \mid \text{Technology}) \cdot P(\text{Technology})}{P(X)} = \frac{0.0001 \times 0.4}{0.07204} \approx 0.001
   ]

Interpretation:

The document $X=\text{"apple banana orange"}$ is overwhelmingly more likely to belong to the Fruits topic ($P(\text{Fruits}\mid X)\approx 0.999$).

---

2. Variational Autoencoders (VAEs)

Problem:

You observe a datapoint $X=3.0$, and you're inferring the latent variable $Z$ in a VAE. The model assumes:
- $Z\sim \mathcal{N}(0,1)$ (prior).
- $X\mid Z\sim \mathcal{N}(Z,{0.5}^2)$ (likelihood).

Generative Model Components:

1. Prior ($P(Z)$):
   [
   P(Z) = \frac{1}{\sqrt{2\pi}} e^{-Z2 / 2}
   ]
2. Likelihood ($P(X\mid Z)$):
   [
   P(X \mid Z) = \frac{1}{\sqrt{2\pi \cdot 0.5^2}} e^{-(X - Z)^2 / (2 \cdot 0.5^2)}
   ]
3. Evidence ($P(X)$):
   Marginalize over $Z$: $P(X)=\int P(X\mid Z)P(Z)dZ$. (In practice, approximated using variational inference.)

Applying Bayes' Theorem:

We want $P(Z\mid X=3.0)$.

1. Posterior ($P(Z\mid X)$):
   [
   P(Z \mid X) \propto P(X \mid Z) \cdot P(Z)
   ]

2. Compute unnormalized posterior:
   Substitute likelihood and prior into the formula:
   [
   P(Z \mid X = 3.0) \propto \left(\frac{1}{\sqrt{2\pi \cdot 0.5^2}} e^{-(3.0 - Z)^2 / (2 \cdot 0.5^2)}\right) \cdot \left(\frac{1}{\sqrt{2\pi}} e^{-Z2 / 2}\right)
   ]

3. Approximation:
   Because $P(X)$ is challenging to compute exactly, VAEs approximate $P(Z\mid X)$ using a learned variational distribution $q(Z\mid X)$.

Interpretation:

The posterior $P(Z\mid X)$ explains which latent variables $Z$ are most likely to have generated the observation $X=3.0$.

---

3. Gaussian Mixture Models (GMMs)

Problem:

You observe a data point $X=4.5$ and need to infer which component $Z$ of a GMM generated it. Suppose:
- $Z\in \{1,2\}$ (two Gaussian components).
- $P(Z=1)=0.7,P(Z=2)=0.3$ (prior).
- $X\mid Z=1\sim \mathcal{N}(5,1^2),X\mid Z=2\sim \mathcal{N}(3,1^2)$ (likelihood).

Applying Bayes' Theorem:

1. Compute likelihoods:
2. $P(X=4.5\mid Z=1)=\frac{1}{\sqrt{2\pi \cdot 1^2}}{e}^{-(4.5-5{)}^2/2}\approx 0.352$

3. $P(X=4.5\mid Z=2)=\frac{1}{\sqrt{2\pi \cdot 1^2}}{e}^{-(4.5-3{)}^2/2}\approx 0.043$

4. Compute posterior probabilities:

5. Normalizing constant:
   [
   P(X) = P(X \mid Z = 1) \cdot P(Z = 1) + P(X \mid Z = 2) \cdot P(Z = 2)
   ]
   [
   P(X) = (0.352 \cdot 0.7) + (0.043 \cdot 0.3) = 0.2464 + 0.0129 = 0.2593
   ]

6. Posterior for $Z=1$:
   [
   P(Z = 1 \mid X = 4.5) = \frac{P(X \mid Z = 1) \cdot P(Z = 1)}{P(X)} = \frac{0.352 \cdot 0.7}{0.2593} \approx 0.951
   ]

7. Posterior for $Z=2$:
   [
   P(Z = 2 \mid X = 4.5) = \frac{P(X \mid Z = 2) \cdot P(Z = 2)}{P(X)} = \frac{0.043 \cdot 0.3}{0.2593} \approx 0.049
   ]

Interpretation:

The observation $X=4.5$ is most likely generated by component $Z=1$ with a posterior probability of approximately 95.1%.

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In the given example, the likelihood $P(X\mid Z)$ is calculated based on the probability of individual words occurring in a document, assuming the words are independent of each other given the topic $Z$. This is a simplifying assumption often used in models like Latent Dirichlet Allocation (LDA).

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Steps to Calculate the Likelihood:

1. Assumption of Independence:
2. The model assumes that the probability of a document $X$ (e.g., "apple banana orange") given a topic $Z$ can be decomposed into the product of the probabilities of each individual word in the document:
   [
   P(X \mid Z) = P(\text{"apple"} \mid Z) \cdot P(\text{"banana"} \mid Z) \cdot P(\text{"orange"} \mid Z)
   ]

3. This is called the bag-of-words assumption, which disregards word order and considers only word frequencies.

4. Word Probabilities:

5. The probabilities $P(\text{word}\mid Z)$ are learned during model training. For example:

   • For topic $Z=\text{Fruits}$:
     [
     P(\text{"apple"} \mid \text{Fruits}) = 0.5, \quad P(\text{"banana"} \mid \text{Fruits}) = 0.4, \quad P(\text{"orange"} \mid \text{Fruits}) = 0.6
     ]
   • For topic $Z=\text{Technology}$:
     [
     P(\text{"apple"} \mid \text{Technology}) = 0.1, \quad P(\text{"banana"} \mid \text{Technology}) = 0.05, \quad P(\text{"orange"} \mid \text{Technology}) = 0.02
     ]

6. Compute the Likelihood:

7. For topic $Z=\text{Fruits}$:
   [
   P(\text{"apple banana orange"} \mid \text{Fruits}) = P(\text{"apple"} \mid \text{Fruits}) \cdot P(\text{"banana"} \mid \text{Fruits}) \cdot P(\text{"orange"} \mid \text{Fruits})
   ]
   [
   = 0.5 \cdot 0.4 \cdot 0.6 = 0.12
   ]
8. For topic $Z=\text{Technology}$:
   [
   P(\text{"apple banana orange"} \mid \text{Technology}) = P(\text{"apple"} \mid \text{Technology}) \cdot P(\text{"banana"} \mid \text{Technology}) \cdot P(\text{"orange"} \mid \text{Technology})
   ]
   [
   = 0.1 \cdot 0.05 \cdot 0.02 = 0.0001
   ]

---

Why This Works:

The likelihood $P(X\mid Z)$ is a measure of how well the topic $Z$ explains the document $X$. By assuming independence between words and knowing the word probabilities $P(\text{word}\mid Z)$ for each topic, we calculate how probable it is to observe the given document $X$ under the topic $Z$.

This approach simplifies the computation but may lose some information about word relationships (e.g., word order).

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Yes, the word probabilities $P(\text{word}\mid Z)$ are not arbitrary assumptions but are learned during the training phase of the model. Here's how this works in detail:

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How Word Probabilities $P(\text{word}\mid Z)$ Are Learned

1. Data Preparation:
2. The model is provided with a large collection of documents.

3. Each document is associated with one or more topics (in supervised models) or the topics are inferred (in unsupervised models like LDA).

4. Objective:

5. The goal is to compute the conditional probability of each word appearing in a document, given the topic ($Z$).

6. For example, for the topic "Fruits," the probability $P(\text{"apple"}\mid \text{"Fruits"})$ represents how likely the word "apple" is to appear in documents dominated by the "Fruits" topic.

7. Training Process:

8. Using algorithms like Expectation-Maximization (EM) or variational inference, the model iteratively:

   1. Assigns probabilities for topics to documents.
   2. Updates word distributions for each topic based on the assigned probabilities.

9. Learned Probabilities:

10. After training, the word-topic probabilities are estimated from the frequency of words in documents related to a particular topic.
11. For example, if "apple" appears frequently in documents labeled or inferred as "Fruits," $P(\text{"apple"}\mid \text{"Fruits"})$ will be high.

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Why Use These Probabilities?

1. Model Simplicity:
2. $P(\text{word}\mid Z)$ is a compact way to represent the characteristics of a topic. Each topic is essentially a distribution over words.
3. Generative Assumption:
4. Generative models assume that words in a document are sampled from the topic's word distribution. For example:
   • If $Z=\text{"Fruits"}$, the words in a document are sampled based on $P(\text{"word"}\mid \text{"Fruits"})$.

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Are the Given Probabilities Assumed in the Example?

For the example provided, the probabilities ($P(\text{"apple"}\mid \text{"Fruits"})=0.5,P(\text{"banana"}\mid \text{"Fruits"})=0.4,P(\text{"orange"}\mid \text{"Fruits"})=0.6$) are example values to demonstrate the process. In a real-world scenario:
- These probabilities are learned during model training from a corpus of documents.
- Once learned, they are fixed and used to compute the likelihood for new documents.

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Summary

• Assumed in Example: The probabilities are given as illustrative values for simplicity.
• Learned in Practice: These probabilities are calculated during training based on observed word frequencies in topic-specific documents.