User: list of common continuous prob distributions and their density functions
Assistant: Here is a list of common continuous probability distributions along with their probability density functions (pdf):

### 1. Uniform Distribution \( \text{Uniform}(a,b) \)
- **Support:** \( x \in [a,b] \)
- **PDF:** 
\[
f(x) = \frac{1}{b - a}, \quad a \leq x \leq b
\]

---

### 2. Normal (Gaussian) Distribution \( \mathcal{N}(\mu, \sigma^2) \)
- **Support:** \( x \in (-\infty, \infty) \)
- **PDF:** 
\[
f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)
\]

---

### 3. Exponential Distribution \( \text{Exp}(\lambda) \)
- **Support:** \( x \geq 0 \)
- **PDF:** 
\[
f(x) = \lambda e^{-\lambda x}
\]

---

### 4. Gamma Distribution \( \Gamma(\alpha, \beta) \) (shape \(\alpha > 0\), rate \(\beta > 0\))
- **Support:** \( x \geq 0 \)
- **PDF:**
\[
f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}
\]

---

### 5. Beta Distribution \( \text{Beta}(\alpha, \beta) \)
- **Support:** \( x \in [0,1] \)
- **PDF:** 
\[
f(x) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}
\]
where \( B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} \)

---

### 6. Chi-Square Distribution (\(\chi^2_k\)) — Special case of Gamma with \(\alpha = k/2, \beta = 1/2\)
- **Support:** \( x \geq 0 \)
- **PDF:**
\[
f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{\frac{k}{2} - 1} e^{-x/2}
\]

---

### 7. Student's t-Distribution \( t_\nu \)
- **Support:** \( x \in (-\infty, \infty) \)
- **PDF:**
\[
f(x) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi} \; \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu + 1}{2}}
\]

---

### 8. F-Distribution \( F(d_1, d_2) \)
- **Support:** \( x \geq 0 \)
- **PDF:**
\[
f(x) = \frac{1}{B\left(\frac{d_1}{2}, \frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{d_1/2} x^{\frac{d_1}{2} - 1} \left(1 + \frac{d_1}{d_2} x\right)^{-\frac{d_1 + d_2}{2}}
\]

---

### 9. Weibull Distribution \( \text{Weibull}(k, \lambda) \)
- **Support:** \( x \geq 0 \)
- **PDF:**
\[
f(x) = \frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}
\]

---

### 10. Log-Normal Distribution \( \text{LogNormal}(\mu, \sigma) \)
- **Support:** \( x > 0 \)
- **PDF:**
\[
f(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2 \sigma^2}\right)
\]

---

If you want pdfs for more specialized distributions or discrete distributions, just let me know!
User: definition of fisher information
Assistant: The **Fisher Information** measures the amount of information that an observable random variable \(X\) carries about an unknown parameter \(\theta\) upon which the probability of \(X\) depends. It quantifies the sensitivity of the likelihood function to changes in the parameter.

### Formal Definition

Suppose \(X\) is a random variable with probability density function (or probability mass function) \( f(x; \theta) \), where \(\theta\) is a parameter (scalar or vector). The Fisher information \( I(\theta) \) is defined as:

\[
I(\theta) = \mathbb{E}_\theta \left[ \left( \frac{\partial}{\partial \theta} \log f(X; \theta) \right)^2 \right]
\]

where the expectation is taken with respect to the distribution of \(X\) given \(\theta\).

Equivalently, under regularity conditions (allowing exchange of derivative and integral), Fisher information can be expressed as:

\[
I(\theta) = - \mathbb{E}_\theta \left[ \frac{\partial^2}{\partial \theta^2} \log f(X; \theta) \right]
\]

### Intuition

- \(\frac{\partial}{\partial \theta} \log f(X; \theta)\) is called the **score function**, representing the gradient of the log-likelihood.
- The Fisher information is the variance of the score, showing how much the log-likelihood changes with \(\theta\).
- Higher Fisher information means more precision in estimating \(\theta\) from data.

### Multivariate Case

If \(\theta\) is a vector parameter, the Fisher information becomes a matrix:

\[
[I(\theta)]_{ij} = \mathbb{E}_\theta \left[ \frac{\partial}{\partial \theta_i} \log f(X; \theta) \cdot \frac{\partial}{\partial \theta_j} \log f(X; \theta) \right]
\]

---

**Summary:**

\[
\boxed{
I(\theta) = \mathbb{E}_\theta \left[ \left(\frac{\partial}{\partial \theta} \log f(X; \theta) \right)^2 \right] = -\mathbb{E}_\theta \left[ \frac{\partial^2}{\partial \theta^2} \log f(X; \theta) \right]
}
\]
User: exp and variance of bernoulli
Assistant: For a Bernoulli random variable \( X \sim \text{Bern}(p) \), where \( p = P(X=1) \) and \( 1 - p = P(X=0) \):

### Expectation (Mean)
\[
\mathbb{E}[X] = p
\]

### Variance
\[
\mathrm{Var}(X) = p(1 - p)
\]

---

**Summary:**

\[
\boxed{
\begin{cases}
\mathbb{E}[X] = p \\
\mathrm{Var}(X) = p(1 - p)
\end{cases}
}
\]
User: exp and variance of binomial
Assistant: For a Binomial random variable \( X \sim \text{Binomial}(n, p) \), where \( n \) is the number of independent trials and \( p \) is the success probability in each trial:

### Expectation (Mean)
\[
\mathbb{E}[X] = n p
\]

### Variance
\[
\mathrm{Var}(X) = n p (1 - p)
\]

---

**Summary:**

\[
\boxed{
\begin{cases}
\mathbb{E}[X] = n p \\
\mathrm{Var}(X) = n p (1 - p)
\end{cases}
}
\]
User: exp and var of poisson
Assistant: For a Poisson random variable \( X \sim \text{Poisson}(\lambda) \), where \(\lambda > 0\) is the rate (or mean number of events):

### Expectation (Mean)
\[
\mathbb{E}[X] = \lambda
\]

### Variance
\[
\mathrm{Var}(X) = \lambda
\]

---

**Summary:**

\[
\boxed{
\begin{cases}
\mathbb{E}[X] = \lambda \\
\mathrm{Var}(X) = \lambda
\end{cases}
}
\]
User: exp and var geometric
Assistant: For a Geometric random variable \( X \), there are two common definitions depending on the parameterization:

---

### Definition 1: Number of **failures before** the first success

- Support: \( X = 0, 1, 2, \dots \)
- Parameter: Success probability \( p \in (0, 1) \)
- PMF: 
\[
P(X = k) = (1-p)^k p, \quad k = 0,1,2,\dots
\]

**Expectation and Variance:**
\[
\mathbb{E}[X] = \frac{1-p}{p}
\]
\[
\mathrm{Var}(X) = \frac{1-p}{p^2}
\]

---

### Definition 2: Number of **trials up to and including** the first success

- Support: \( X = 1, 2, 3, \dots \)
- Parameter: Success probability \( p \in (0, 1) \)
- PMF:
\[
P(X = k) = (1-p)^{k-1} p, \quad k = 1,2,3,\dots
\]

**Expectation and Variance:**
\[
\mathbb{E}[X] = \frac{1}{p}
\]
\[
\mathrm{Var}(X) = \frac{1-p}{p^2}
\]

---

**Summary:**

| Definition                          | Support           | \( \mathbb{E}[X] \)      | \( \mathrm{Var}(X) \)        |
|-----------------------------------|-------------------|--------------------------|------------------------------|
| Failures before first success      | \( 0,1,2,\dots \) | \( \frac{1-p}{p} \)      | \( \frac{1-p}{p^2} \)        |
| Trials until first success          | \( 1,2,3,\dots \) | \( \frac{1}{p} \)        | \( \frac{1-p}{p^2} \)        |

---

Let me know if you want details or derivations!
User: exp and var neg binom
Assistant: For a Negative Binomial random variable \( X \sim \text{NegBin}(r, p) \), there are two common parameterizations. Here, we consider the common one where:

- \(r > 0\) is the number of **successes** to be observed,
- \(p \in (0,1)\) is the probability of **success** on each trial,
- \(X\) counts the number of **failures** before the \(r\)-th success.

---

### Negative Binomial \(X =\) Number of failures before the \(r\)-th success

- Support: \(X = 0, 1, 2, \ldots\)

#### Expectation (Mean):
\[
\mathbb{E}[X] = \frac{r(1-p)}{p}
\]

#### Variance:
\[
\mathrm{Var}(X) = \frac{r(1-p)}{p^2}
\]

---

### Alternative parameterization: \(Y = X + r\) is the number of trials until the \(r\)-th success.

- Support: \(Y = r, r+1, r+2, \ldots\)

#### Expectation:
\[
\mathbb{E}[Y] = \frac{r}{p}
\]

#### Variance:
\[
\mathrm{Var}(Y) = \frac{r(1-p)}{p^2}
\]

---

### Notes:

- When \(r=1\), the Negative Binomial reduces to the Geometric distribution (failures before 1st success).
- The Negative Binomial describes the sum of \(r\) independent Geometric random variables (failures before success).
  
---

### Summary:

\[
\boxed{
\begin{cases}
X \sim \text{NegBin}(r,p): \text{failures before } r\text{-th success} \\
\mathbb{E}[X] = \dfrac{r(1-p)}{p} \\
\mathrm{Var}(X) = \dfrac{r(1-p)}{p^2}
\end{cases}
}
\]

If you want formulas or details for alternative parameterizations, let me know!
User: exp and var uniform continuous
Assistant: For a continuous random variable \( X \sim \text{Uniform}(a,b) \) with \( a < b \):

---

### Expectation (Mean)
\[
\mathbb{E}[X] = \frac{a+b}{2}
\]

---

### Variance
\[
\mathrm{Var}(X) = \frac{(b - a)^2}{12}
\]

---

### Summary:
\[
\boxed{
\begin{cases}
\mathbb{E}[X] = \frac{a+b}{2} \\
\mathrm{Var}(X) = \frac{(b - a)^2}{12}
\end{cases}
}
\]
User: exp and var exponential
Assistant: For a continuous random variable \( X \sim \text{Exponential}(\lambda) \) with rate parameter \(\lambda > 0\), defined on \( x \geq 0 \):

---

### Expectation (Mean)
\[
\mathbb{E}[X] = \frac{1}{\lambda}
\]

---

### Variance
\[
\mathrm{Var}(X) = \frac{1}{\lambda^2}
\]

---

### Summary:
\[
\boxed{
\begin{cases}
\mathbb{E}[X] = \frac{1}{\lambda} \\
\mathrm{Var}(X) = \frac{1}{\lambda^2}
\end{cases}
}
\]
User: now gamma
Assistant: For a Gamma random variable \( X \sim \Gamma(\alpha, \beta) \), where:
- \(\alpha > 0\) is the **shape** parameter,
- \(\beta > 0\) is the **rate** parameter (sometimes denoted as \(\theta = 1/\beta\) called scale),

and the PDF is:
\[
f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}, \quad x \geq 0,
\]

---

### Expectation (Mean)
\[
\mathbb{E}[X] = \frac{\alpha}{\beta}
\]

---

### Variance
\[
\mathrm{Var}(X) = \frac{\alpha}{\beta^2}
\]

---

### Notes:
- Sometimes the Gamma is parameterized using a **scale** parameter \(\theta = 1/\beta\) instead of rate:
  \[
  f(x) = \frac{1}{\Gamma(\alpha) \theta^\alpha} x^{\alpha - 1} e^{-x/\theta}
  \]
  In that case,
  \[
  \mathbb{E}[X] = \alpha \theta, \quad \mathrm{Var}(X) = \alpha \theta^2.
  \]

---

### Summary (rate parameterization):
\[
\boxed{
\begin{cases}
\mathbb{E}[X] = \frac{\alpha}{\beta} \\
\mathrm{Var}(X) = \frac{\alpha}{\beta^2}
\end{cases}
}
\]

Let me know if you want details on other parameterizations or derivations!
User: now beta
Assistant: For a Beta random variable \( X \sim \text{Beta}(\alpha, \beta) \) with parameters \(\alpha > 0\) and \(\beta > 0\), defined on the interval \( [0, 1] \), the probability density function is:

\[
f(x) = \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \quad 0 \leq x \leq 1,
\]
where \( B(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} \) is the Beta function.

---

### Expectation (Mean)
\[
\mathbb{E}[X] = \frac{\alpha}{\alpha + \beta}
\]

---

### Variance
\[
\mathrm{Var}(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}
\]

---

### Summary:
\[
\boxed{
\begin{cases}
\mathbb{E}[X] = \frac{\alpha}{\alpha + \beta} \\
\mathrm{Var}(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}
\end{cases}
}
\]

Let me know if you want more details or moments of other distributions!
User: guidelines for determining the rejection region of a hypothesis test given alpha
Assistant: When performing a hypothesis test at significance level \(\alpha\), determining the **rejection region** involves deciding which values of the test statistic will lead you to reject the null hypothesis \(H_0\).

Here are the general **guidelines** to determine the rejection region:

---

### 1. **Identify the Hypotheses and Test Type**

- **Null hypothesis:** \(H_0\)
- **Alternative hypothesis:** \(H_a\) (can be one-sided or two-sided)
  
Types of tests (assume test statistic \(T\)):

- **Right-tailed test:** \(H_a: \theta > \theta_0\)
- **Left-tailed test:** \(H_a: \theta < \theta_0\)
- **Two-tailed test:** \(H_a: \theta \neq \theta_0\)

---

### 2. **Specify the Significance Level**

\[
\alpha = P(\text{Type I error}) = P(\text{reject } H_0 \mid H_0 \text{ true})
\]

Common values: 0.05, 0.01, 0.10

---

### 3. **Find the Distribution of the Test Statistic under \(H_0\)**

Determine the sampling distribution of \(T\) assuming \(H_0\) is true. It could be:

- Normal,
- t-distribution,
- Chi-square,
- F-distribution,
- Others.

---

### 4. **Determine Critical Value(s) Corresponding to \(\alpha\)**

- **Right-tailed test:** Find \(c\) such that:

\[
P(T > c \mid H_0) = \alpha
\]

Rejection region:

\[
\{ T > c \}
\]

- **Left-tailed test:** Find \(c\) such that:

\[
P(T < c \mid H_0) = \alpha
\]

Rejection region:

\[
\{ T < c \}
\]

- **Two-tailed test:** Find critical values \(c_1, c_2\) such that:

\[
P(T < c_1 \mid H_0) = \frac{\alpha}{2}, \quad P(T > c_2 \mid H_0) = \frac{\alpha}{2}
\]

Rejection region:

\[
\{ T < c_1 \} \cup \{ T > c_2 \}
\]

---

### 5. **Express the Rejection Region**

Write explicitly the set of test statistic values where you reject \(H_0\).

---

### 6. **Use Tables or Statistical Software to Find Critical Values**

- For common distributions, critical values (quantiles) are tabulated.
- Alternatively, use software functions like `norm.ppf()`, `t.ppf()`, `chi2.ppf()`, etc.

---

### **Example Summary:**

| Test Type        | Rejection Region                      | Critical Value(s)                     |
|------------------|-------------------------------------|-------------------------------------|
| Right-tailed     | \( T > c \)                        | \( c = F_T^{-1}(1-\alpha) \)        |
| Left-tailed      | \( T < c \)                        | \( c = F_T^{-1}(\alpha) \)           |
| Two-tailed      | \( T < c_1 \) or \( T > c_2 \)    | \( c_1 = F_T^{-1}(\alpha/2), \quad c_2 = F_T^{-1}(1 - \alpha/2) \) |

where \(F_T^{-1}\) denotes the quantile (inverse CDF) function of the test statistic \(T\) under the null hypothesis.

---

### **Additional Tips**

- Be clear about the direction of the alternative hypothesis.
- Check assumptions to ensure the test statistic distribution is valid.
- For discrete test statistics, rejection regions may not exactly match \(\alpha\) but should be as close as possible without exceeding \(\alpha\).
- For complex tests, consider using p-values rather than fixed rejection regions.

---

If you'd like, I can provide examples for specific hypothesis tests!
User: definition of bias and consistency
Assistant: Here are the definitions of **bias** and **consistency** in the context of statistical estimation:

---

### 1. Bias

**Bias** of an estimator \(\hat{\theta}\) of a parameter \(\theta\) measures the difference between the estimator's expected value and the true parameter value.

\[
\boxed{
\text{Bias}(\hat{\theta}) = \mathbb{E}[\hat{\theta}] - \theta
}
\]

- **Unbiased estimator:** \(\mathbb{E}[\hat{\theta}] = \theta\), so \(\text{Bias}(\hat{\theta}) = 0\).
- **Biased estimator:** If \(\text{Bias}(\hat{\theta}) \neq 0\).

**Interpretation:**  
Bias indicates systematic error in estimation. An unbiased estimator, on average, hits the true parameter value; a biased estimator systematically underestimates or overestimates.

---

### 2. Consistency

An estimator \(\hat{\theta}_n\) (based on a sample of size \(n\)) is **consistent** for \(\theta\) if it converges in probability to the true parameter as the sample size grows:

\[
\boxed{
\hat{\theta}_n \xrightarrow{p} \theta \quad \text{as } n \to \infty
}
\]
which means that for any \(\varepsilon > 0\):

\[
\lim_{n \to \infty} P(|\hat{\theta}_n - \theta| > \varepsilon) = 0
\]

- Roughly, the estimator becomes arbitrarily close to \(\theta\) with high probability as more data is collected.

---

### **Summary:**

| Property   | Definition                                      | Interpretation                        |
|------------|------------------------------------------------|-------------------------------------|
| **Bias**  | \(\mathbb{E}[\hat{\theta}] - \theta\)          | Systematic deviation from true value |
| **Consistency** | \(\hat{\theta}_n \xrightarrow{p} \theta\) as \(n \to \infty\) | Estimator converges to true value with large samples |

---

If you want, I can also explain related concepts like efficiency or mean squared error!
User: uniformly most powerful test criteria
Assistant: *Uniformly Most Powerful (UMP) Test:**

A **Uniformly Most Powerful (UMP)** test is a hypothesis test that maximizes the power (probability of correctly rejecting the null hypothesis when the alternative is true) **for all parameter values** in the alternative hypothesis class, subject to a fixed size \(\alpha\) (type I error rate).

---

### Criteria for UMP Tests:

1. **Definition:**

For testing

\[
H_0: \theta = \theta_0 \quad \text{vs} \quad H_1: \theta \in \Theta_1,
\]

a test \(\phi^*\) is **UMP at level \(\alpha\)** if:

- \(\mathbb{E}_{\theta_0}[\phi^*(X)] = \alpha\) (test has size \(\alpha\)),
- For **every** other test \(\phi\) of size \(\leq \alpha\), and for **every** \(\theta \in \Theta_1\),
\[
\mathbb{E}_{\theta}[\phi^*(X)] \geq \mathbb{E}_{\theta}[\phi(X)].
\]

In words, \(\phi^*\) has the greatest power uniformly over all \(\theta \in \Theta_1\).

---

2. **Neyman-Pearson Lemma (Basis for UMP for Simple vs Simple Tests):**

- For testing \(H_0: \theta = \theta_0\) vs \(H_1: \theta = \theta_1\),
- The most powerful test rejects \(H_0\) for large values of the likelihood ratio:
\[
\Lambda(x) = \frac{f(x; \theta_1)}{f(x; \theta_0)} > k,
\]
where \(k\) is chosen to satisfy size \(\alpha\).

---

3. **Extension to Composite Alternatives (Conditions for UMP Existence):**

- UMP tests often exist if the family of distributions has a **Monotone Likelihood Ratio (MLR)** property in a statistic \(T(X)\).
- The test statistic \(T(X)\) satisfies that for \(\theta_1 > \theta_0\),
\[
\frac{f_{T}(t; \theta_1)}{f_{T}(t; \theta_0)} \text{ is non-decreasing in } t,
\]
where \(f_T\) is the density of \(T\).

- Under MLR, a test that rejects \(H_0\) if \(T(X) > c\) for a certain cutoff \(c\) is UMP for testing:

\[
H_0: \theta \leq \theta_0 \quad \text{vs} \quad H_1: \theta > \theta_0.
\]

---

4. **UMP Tests May Not Exist:**

- In many cases (especially two-sided tests or multidimensional parameters), no UMP test exists.
- One resorts to:
  - UMP unbiased tests,
  - Likelihood ratio tests,
  - Other optimality criteria.

---

### **Summary**

| Criterion                 | Details                                                                                      |
|---------------------------|----------------------------------------------------------------------------------------------|
| **UMP test**              | Maximizes power uniformly over all parameter values in \(H_1\) at fixed significance level \(\alpha\). |
| **Exists when**           | A monotone likelihood ratio (MLR) condition holds in a scalar test statistic \(T\).          |
| **Test form**             | Reject \(H_0\) if \(T(X)\) exceeds a critical value \(c\).                                  |
| **Neyman-Pearson Lemma**  | Provides Most Powerful test for simple hypotheses; basis for UMP under MLR condition.        |
| **Often does not exist**  | For two-sided or multidimensional composite alternatives.                                   |

---

If you want, I can provide examples or explain MLR
User: posterior predictive distribution
Assistant: The **posterior predictive distribution** is a key concept in Bayesian statistics. It represents the distribution of a future (new) observation \( \tilde{X} \) given the observed data \( X \).

---

### Setup

- You have observed data: \( X = x \).
- You have a model with parameter \( \theta \).
- After observing data, you obtain the **posterior distribution**:
\[
p(\theta \mid x)
\]
which expresses uncertainty about \(\theta\) after seeing data.

---

### Definition of Posterior Predictive Distribution

The posterior predictive distribution for a new observation \( \tilde{X} \) is the **distribution of \(\tilde{X}\) averaged over the posterior distribution of \(\theta\)**:

\[
p(\tilde{x} \mid x) = \int p(\tilde{x} \mid \theta) \, p(\theta \mid x) \, d\theta
\]

- Here, \( p(\tilde{x} \mid \theta) \) is the likelihood or sampling distribution for the new data given parameter \(\theta\).
- The integral averages over all possible \(\theta\), weighted by how plausible each \(\theta\) is given the observed data.

---

### Interpretation

- This distribution predicts what future data might look like, incorporating both the uncertainty in \(\theta\) and inherent randomness.
- It is a **mixture** of likelihoods weighted by the posterior on \(\theta\).

---

### Example

Suppose:

- \( X \mid \theta \sim \mathcal{N}(\theta, \sigma^2) \),
- Prior and data give posterior \( p(\theta \mid x) \),
- Then:
\[
p(\tilde{x} \mid x) = \int \mathcal{N}(\tilde{x} \mid \theta, \sigma^2) \, p(\theta \mid x) \, d\theta
\]

If the posterior is also normal (due to conjugacy), this integral often results in a normal predictive distribution with mean equal to the posterior mean and variance equal to **data variance + posterior variance**.

---

### Summary

\[
\boxed{
\text{Posterior Predictive:} \quad p(\tilde{x} \mid x) = \int p(\tilde{x} \mid \theta) \, p(\theta \mid x) \, d\theta
}
\]

- Combines uncertainty about parameters and inherent randomness of data.
- Used for prediction, model checking, Bayesian decision theory.

---

If you want, I can provide an explicit example or derivation for a specific model!
User: what is meant by canonical form of exponential families
Assistant: The **canonical form of an exponential family** refers to a specific representation of a family of probability distributions that expresses their probability density function (pdf) or probability mass function (pmf) in a standardized, factorized form involving:

- A **natural parameter** (also called canonical parameter),
- A **sufficient statistic**,
- A **log-partition function** (normalizer),
- And a base measure.

---

### Formal Definition

A probability distribution belongs to the **exponential family** if its pdf (or pmf) with respect to some base measure \(\nu\) can be written as:

\[
f_X(x; \theta) = h(x) \exp\Big( \eta(\theta)^\top T(x) - A(\theta) \Big),
\]

where:

- \(x\) is the data,
- \(\theta\) is the original parameter (possibly multidimensional),
- \(\eta(\theta)\) is the **natural (canonical) parameter**, a function mapping \(\theta\) to the canonical parameter space,
- \(T(x)\) is the **sufficient statistic** (can be vector-valued),
- \(A(\theta)\) is the **log-partition function** (also called cumulant generating function), ensuring the density integrates to 1,
- \(h(x)\) is the **base measure** (or carrier measure), independent of \(\theta\).

---

### Canonical (Natural) Form

The form above is called **canonical** when the parameter is expressed directly as the natural parameter, often written simply as \(\eta\), giving:

\[
f_X(x; \eta) = h(x) \exp\big( \eta^\top T(x) - A(\eta) \big).
\]

Here:

- The family is parametrized by \(\eta\) directly,
- \(A(\eta)\) is convex in \(\eta\),
- The sufficient statistic \(T(x)\) is fixed,
- This representation reveals important properties such as conjugacy, moments, and convexity.

---

### Why Canonical Form Matters

- **Simplifies analysis:** Many properties (e.g., moments, variance) are easier to derive,
- **Key to conjugate priors:** Prior distributions that are conjugate to the model are often expressed in terms of the canonical parameter,
- **Provides natural parametrization:** \(\eta\) is often more mathematically convenient than the original \(\theta\),
- **Unifies many distributions:** Examples like Gaussian, Bernoulli, Poisson, Gamma, Beta can be expressed as exponential families in canonical form.

---

### Example: Bernoulli Distribution

Original parameter: \(p \in (0,1)\) — probability of success.

PDF:
\[
f(x; p) = p^x (1-p)^{1-x}, \quad x \in \{0,1\}.
\]

Rewrite in canonical form:

- Sufficient statistic: \(T(x) = x\),
- Natural parameter: \(\eta = \log \frac{p}{1-p}\) (log-odds),
- Base measure: \(h(x) = 1\),
- Log-partition function:
\[
A(\eta) = \log(1 + e^{\eta}),
\]
since:
\[
f(x; \eta) = \exp\left( \eta x - \log(1 + e^{\eta}) \right).
\]

---

### Summary

\[
\boxed{
\text{Canonical form of exponential family:} \quad f(x; \eta) = h(x) \exp\left( \eta^\top T(x) - A(\eta) \right)
}
\]

- \(\eta\): natural (canonical) parameter,
- \(T(x)\): sufficient statistic,
- \(A(\eta)\): log-partition function normalizing the distribution,
- \(h(x)\): base measure, independent of \(\eta\).

---

If you'd like, I can provide canonical forms for other distributions or examples!
User: list of mcmc schemes
Assistant: Here is a list of common **Markov Chain Monte Carlo (MCMC) schemes**, techniques used to sample from complex probability distributions:

---

### 1. **Metropolis-Hastings (MH) Algorithm**
- Proposes candidate moves from a proposal distribution \( q(x' \mid x) \).
- Accepts or rejects the candidate based on an acceptance probability to ensure detailed balance.
- General-purpose and widely used.

---

### 2. **Metropolis Algorithm**
- Special case of MH with symmetric proposal \(q(x' \mid x) = q(x \mid x')\).
- Acceptance probability simplifies.

---

### 3. **Gibbs Sampling**
- Samples each variable (or block of variables) conditionally from its full conditional distribution.
- Requires the ability to sample directly from conditionals.

---

### 4. **Blocked Gibbs Sampling**
- Generalization of Gibbs, where groups of variables are updated jointly.

---

### 5. **Hamiltonian Monte Carlo (HMC) / Hybrid Monte Carlo**
- Uses gradients of the log-target density.
- Introduces auxiliary momentum variables and simulates Hamiltonian dynamics.
- Efficient for high-dimensional continuous distributions.
- Requires gradients and smooth target densities.

---

### 6. **No-U-Turn Sampler (NUTS)**
- An adaptive variant of HMC.
- Automatically tunes path length to avoid manual parameter choice.

---

### 7. **Slice Sampling**
- Introduces an auxiliary variable to sample uniformly from the region under the density curve.
- Can adapt to the shape of the distribution without tuning step sizes.

---

### 8. **Adaptive MCMC**
- The proposal distribution adapts based on the past samples to improve efficiency.
- Examples: Adaptive Metropolis, Adaptive Metropolis-within-Gibbs.

---

### 9. **Reversible Jump MCMC (RJMCMC)**
- Allows jumps between parameter spaces of different dimensions.
- Used for model selection and variable dimension problems.

---

### 10. **Parallel Tempering (Replica Exchange MCMC)**
- Runs multiple chains at different “temperatures” to explore multimodal distributions.
- Chains at higher temperature explore more broadly; swaps between chains allow better mixing.

---

### 11. **Gibbs-within-Metropolis**
- Combines Gibbs sampling for some variables and Metropolis updates for others.

---

### 12. **Delayed Rejection**
- A variant of MH that proposes a second candidate if the first is rejected, potentially improving efficiency.

---

### 13. **Elliptical Slice Sampling**
- Specialized for Gaussian process models.
- Efficiently samples from Gaussian priors conditioned on likelihood.

---

### 14. **Hit-and-Run Sampling**
- Proposes moves in random directions in high-dimensional space.
- Used in constrained or bounded problems.

---

### 15. **Component-wise MH**
- Updates components sequentially using Metropolis steps.

---

### Summary Table:

| MCMC Scheme                | Key Features                                  | Use Cases                          |
|----------------------------|----------------------------------------------|----------------------------------|
| Metropolis-Hastings        | General-purpose, accepts/rejects proposals   | Broad, discrete & continuous     |
| Gibbs Sampling             | Sample from conditionals                      | When conditionals known          |
| Hamiltonian Monte Carlo    | Uses gradients for efficient proposals        | Continuous, high-dim parameters  |
| No-U-Turn Sampler (NUTS)  | Adaptive HMC without tuning path length       | High-dimensional, smooth models  |
| Slice Sampling             | Auxiliary variable, no tuning step size       | When density evaluation easy     |
| Reversible Jump MCMC       | Variable dimension parameter spaces           | Model selection                  |
| Parallel Tempering         | Multiple chains with temperature ladder       | Multimodal distributions         |

---

If you want, I can provide **pseudocode**, more details, or comparisons for any of these methods!
User: efficiency and rao-blackwellisation
Assistant: ## Efficiency (of an estimator)

**Efficiency** quantifies the quality of an estimator relative to the best possible unbiased estimator (usually in terms of variance).

- For an unbiased estimator \(\hat{\theta}\) of a parameter \(\theta\), the **efficiency** is often defined relative to the **Cramér-Rao Lower Bound (CRLB)**:

\[
\text{Efficiency}(\hat{\theta}) = \frac{\text{CRLB}}{\mathrm{Var}(\hat{\theta})}
\]

where the CRLB is the minimum achievable variance by any unbiased estimator.

- Efficiency takes values in \((0,1]\):
  - \(1\) means the estimator achieves the CRLB (called **efficient**),
  - Less than 1 means the estimator is less efficient.

- For two unbiased estimators \(\hat{\theta}_1\) and \(\hat{\theta}_2\), the relative efficiency is often:

\[
\frac{\mathrm{Var}(\hat{\theta}_2)}{\mathrm{Var}(\hat{\theta}_1)}
\]

indicating how many more samples \(\hat{\theta}_2\) would need to match \(\hat{\theta}_1\)’s variance.

---

### Rao-Blackwellisation

**Rao-Blackwellisation** (Rao-Blackwell Theorem) is a procedure to improve an estimator by conditioning it on a **sufficient statistic** without increasing bias, often reducing variance.

---

#### The Rao-Blackwell Theorem:

Given:

- An estimator \(\hat{\theta}\) of \(\theta\) (possibly biased, but usually unbiased),
- A sufficient statistic \(T\) for \(\theta\),

then the **Rao-Blackwellised estimator**

\[
\hat{\theta}^* = \mathbb{E}[\hat{\theta} \mid T]
\]

satisfies:

1. **Unbiasedness is preserved:**

\[
\mathbb{E}[\hat{\theta}^*] = \mathbb{E}[\hat{\theta}] = \theta
\]

(if \(\hat{\theta}\) is unbiased).

2. **Variance is reduced or remains the same:**

\[
\mathrm{Var}(\hat{\theta}^*) \leq \mathrm{Var}(\hat{\theta}).
\]

---

#### Intuition

- By conditioning on the sufficient statistic \(T\), which contains all information about \(\theta\) in the sample,
- you refine the estimator,
- making it a function of \(T\), often greatly reducing variance.

---

#### Practical Use

- Start with a simple estimator (even a crude one),
- Find a sufficient statistic,
- Compute the conditional expectation of the estimator given the sufficient statistic,
- The result is a (usually) improved estimator.

---

### Summary:

| Concept              | Definition / Property                                                                                  |
|----------------------|------------------------------------------------------------------------------------------------------|
| **Efficiency**       | Ratio of minimum achievable variance (CRLB) to actual variance; measures estimator quality            |
| **Rao-Blackwellisation** | Improves estimator by conditioning on sufficient statistic; reduces variance without increasing bias |

---

If you want, I can provide an example applying Rao-Blackwellisation or discuss efficiency in a Bayesian context!
User: score function bayesian
Assistant: In a **Bayesian context**, the term **score function** is less commonly used than in classical (frequentist) inference, but the concept can still be defined and interpreted analogously.

---

### Classical (Frequentist) Score Function Recap

- For a model with parameter \(\theta\) and data \(X\), the **score function** is the gradient of the log-likelihood:

\[
U(\theta) = \frac{\partial}{\partial \theta} \log f(X; \theta).
\]

- It measures how sensitive the likelihood is to changes in \(\theta\).
- The expectation of the score under the true parameter is zero:
\[
\mathbb{E}_\theta[U(\theta)] = 0.
\]

---

### Bayesian "Score Function"

In Bayesian statistics, the parameter \(\theta\) is treated as a random variable with a prior distribution \( p(\theta) \), updated to a posterior \( p(\theta \mid X) \) given data.

The analog of the score function can be considered as the derivative (gradient) of the **log posterior density** with respect to \(\theta\):

\[
S(\theta) := \frac{\partial}{\partial \theta} \log p(\theta \mid X).
\]

Using Bayes’ rule,

\[
\log p(\theta \mid X) = \log f(X \mid \theta) + \log p(\theta) - \log p(X),
\]

where:
- \( f(X \mid \theta) \) is the likelihood,
- \( p(\theta) \) is the prior,
- \( p(X) \) is the marginal likelihood (integral over \(\theta\), constant w.r.t. \(\theta\)).

The gradient is:

\[
S(\theta) = \frac{\partial}{\partial \theta} \log f(X \mid \theta) + \frac{\partial}{\partial \theta} \log p(\theta).
\]

---

### Interpretation:

- The **Bayesian score function** \(S(\theta)\) combines information from the data (likelihood score) and the prior.
- It is used in algorithms like gradient-based MCMC (e.g., Hamiltonian Monte Carlo), variational inference, or for finding the MAP estimate via gradient ascent.
- Unlike the classical score function, in Bayesian inference there is no zero expectation property because \(\theta\) is random and the posterior depends on both prior and likelihood.

---

### Summary

\[
\boxed{
\text{Bayesian Score Function:} \quad S(\theta) = \nabla_\theta \log p(\theta \mid X) = \nabla_\theta \log f(X \mid \theta) + \nabla_\theta \log p(\theta)
}
\]

---

If you want, I can also explain how the Bayesian score relates to posterior modes, or its role in Bayesian information geometry.
User: score function classical
Assistant: In **classical (frequentist) statistics**, the **score function** is a fundamental quantity associated with the likelihood function. It measures the sensitivity of the log-likelihood to changes in the parameter \(\theta\).

---

### Definition

Given a random sample \(X\) with probability density (or mass) function \( f(x; \theta) \), where \(\theta\) is a scalar or vector parameter, the **score function** \( U(\theta) \) is:

\[
\boxed{
U(\theta) = \frac{\partial}{\partial \theta} \log f(X; \theta)
}
\]

- When \(\theta\) is a vector parameter, the score function is the **gradient vector**:
\[
U(\theta) = \nabla_\theta \log f(X; \theta) = \left(\frac{\partial}{\partial \theta_1} \log f(X; \theta), \ldots, \frac{\partial}{\partial \theta_d} \log f(X; \theta)\right)^\top.
\]

---

### Interpretation

- The score function measures how the **log-likelihood changes with \(\theta\)**.
- It points in the direction of steepest ascent of the log-likelihood function.
- It is used in maximum likelihood estimation: the MLE \(\hat{\theta}\) satisfies \(U(\hat{\theta}) = 0\) under regularity conditions (i.e., is a stationary point).

---

### Key Properties

1. **Zero mean under the true parameter:**

\[
\mathbb{E}_{\theta}[U(\theta)] = \int \frac{\partial}{\partial \theta} \log f(x; \theta) f(x;\theta) dx = 0.
\]

2. **Variance and Fisher Information:**

The variance of the score function equals the **Fisher Information**:

\[
I(\theta) = \mathrm{Var}_\theta \big( U(\theta) \big) = \mathbb{E}_\theta\left[ \left( \frac{\partial}{\partial \theta} \log f(X;\theta) \right)^2 \right].
\]

This reflects how much information about \(\theta\) the data carry.

---

### Example

For a single observation from a Normal distribution \(N(\mu, \sigma^2)\), with known \(\sigma^2\), the score with respect to \(\mu\) is:

\[
U(\mu) = \frac{\partial}{\partial \mu} \log f(x; \mu) = \frac{x - \mu}{\sigma^2}.
\]

---

### Summary

| Concept              | Definition                                    |
|----------------------|-----------------------------------------------|
| **Score function**   | \( U(\theta) := \displaystyle \frac{\partial}{\partial \theta} \log f(X; \theta) \) |
| **Interpretation**   | Gradient of log-likelihood, directs increase of likelihood |
| **Property**         | \( \mathbb{E}_{\theta}[U(\theta)] = 0 \)     |
| **Relation**         | Variance of score = Fisher information \(I(\theta)\) |

---

Let me know if you want examples or usage of the score function in inference!
User: def of likelihood function
Assistant: The **likelihood function** is a fundamental concept in statistical inference that quantifies how plausible a particular parameter value is given the observed data.

---

### Definition

Given observed data \( x \) from a random variable \( X \), whose distribution depends on an unknown parameter \(\theta\), the **likelihood function** is defined as:

\[
L(\theta \mid x) = f(x; \theta)
\]

where \( f(x; \theta) \) is the probability density function (pdf) if \(X\) is continuous, or the probability mass function (pmf) if \(X\) is discrete.

---

### Key points:

- **Likelihood as a function of \(\theta\):**  
While \(f(x; \theta)\) is originally a function of \(x\) for fixed \(\theta\), the likelihood treats \(x\) as fixed (observed data) and views the function as depending on the parameter \(\theta\).

- **Not a probability distribution:**  
The likelihood function is **not** a probability distribution over \(\theta\) (unless combined with a prior in Bayesian inference). It need not integrate or sum to 1 over \(\theta\).

---

### Intuition

- The likelihood function measures how probable the observed data \(x\) are for each possible value of \(\theta\).
- It is central to parameter estimation methods such as:
  - **Maximum Likelihood Estimation (MLE):** chooses \(\hat{\theta}\) maximizing \(L(\theta \mid x)\),
  - **Likelihood ratio tests,** confidence intervals, and Bayesian inference (as part of Bayes’ theorem).

---

### Summary

\[
\boxed{
L(\theta \mid x) = f(x; \theta) \quad \text{(viewed as a function of } \theta \text{ with } x \text{ fixed)}
}
\]

---

If you want, I can provide examples with specific distributions or discuss log-likelihood functions!
User: estimator convergence classical statistics
Assistant: In **classical (frequentist) statistics**, **convergence of estimators** refers to the behavior of a sequence of estimators \(\{\hat{\theta}_n\}\) as the sample size \(n \to \infty\). The goal is to understand whether and how \(\hat{\theta}_n\) approaches the true parameter \(\theta\).

---

### Common Modes of Convergence for Estimators

1. **Convergence in Probability (Consistency)**

\[
\hat{\theta}_n \xrightarrow{p} \theta \quad \text{means} \quad \forall \varepsilon > 0, \quad \lim_{n \to \infty} P(|\hat{\theta}_n - \theta| > \varepsilon) = 0.
\]

- An estimator that converges in probability to \(\theta\) is called **consistent**.
- Intuitively, with more data, the estimator gets arbitrarily close to \(\theta\) with high probability.

---

2. **Almost Sure Convergence (Strong Consistency)**

\[
\hat{\theta}_n \xrightarrow{a.s.} \theta \quad \text{means} \quad P\left( \lim_{n \to \infty} \hat{\theta}_n = \theta \right) = 1.
\]

- Stronger than convergence in probability.
- The estimator converges to \(\theta\) with probability 1.
- Often more difficult to prove but provides a stronger guarantee.

---

3. **Convergence in Distribution**

\[
\hat{\theta}_n \xrightarrow{d} \theta,
\]

usually used when \(\theta\) is a constant, this means the distribution of \(\hat{\theta}_n\) converges to a degenerate distribution at \(\theta\).

- Often used in context of the **asymptotic distribution** of scaled estimators, e.g., for normalized estimators:

\[
\sqrt{n}(\hat{\theta}_n - \theta) \xrightarrow{d} \mathcal{N}(0, \sigma^2),
\]

which is central in deriving confidence intervals and hypothesis tests.

---

### Relationships

- Almost sure convergence \(\implies\) convergence in probability \(\implies\) convergence in distribution.
- The reverse implications do **not** generally hold.

---

### Summary

| Mode of Convergence           | Notation                 | Intuition                          |
|------------------------------|--------------------------|----------------------------------|
| **Weak consistency**          | \(\hat{\theta}_n \xrightarrow{p} \theta\)  | Estimates concentrate near \(\theta\) with high probability as \(n \to \infty\) |
| **Strong consistency**        | \(\hat{\theta}_n \xrightarrow{a.s.} \theta\) | Estimates converge to \(\theta\) almost surely          |
| **Asymptotic distribution**   | \(\sqrt{n}(\hat{\theta}_n - \theta) \xrightarrow{d} \mathcal{N}(0, \sigma^2)\) | Distribution of scaled errors tends to normal         |

---

### Example

- The sample mean \(\bar{X}_n\) of i.i.d. observations with mean \(\mu\) satisfies:
  - \(\bar{X}_n \xrightarrow{a.s.} \mu\) (Strong Law of Large Numbers)
  - \(\sqrt{n}(\bar{X}_n - \mu) \xrightarrow{d} \mathcal{N}(0, \sigma^2)\) (Central Limit Theorem)

---

Let me know if you want proofs, examples, or explanation of specific convergence theorems!
User: applications of central limit theorem in classical probability
Assistant: The **Central Limit Theorem (CLT)** is one of the most fundamental results in classical probability and statistics, with numerous theoretical and practical applications. Here are key applications of the CLT:

---

### 1. **Approximating Distributions of Sums and Averages**

- When you sum or average a large number of independent, identically distributed (i.i.d.) random variables (with finite mean and variance), the distribution of the normalized sum/average tends to a Normal distribution.
- This allows you to **approximate complicated distributions** by a normal distribution for sufficiently large sample sizes, regardless of the original distribution.

---

### 2. **Justification for Normal-Based Statistical Procedures**

- Many classical statistical methods (confidence intervals, hypothesis tests) rely on normality assumptions.
- The CLT **justifies using**:
  - \(t\)-tests and \(z\)-tests for means,
  - Confidence intervals based on the sample mean,
  - Approximate normal inference in many parametric and nonparametric methods.
  
Even if the original data is not normally distributed, the sample mean is approximately normal for large samples.

---

### 3. **Construction of Confidence Intervals**

- The CLT provides the basis for constructing **approximate confidence intervals** for population means, proportions (via approximate normality of sample proportions), and other statistics derived from sums or averages.

---

### 4. **Hypothesis Testing**

- Many test statistics (like the sample mean, difference of means, sums) have approximately normal sampling distributions under the null hypothesis due to the CLT.
- This enables calculation of approximate p-values and critical values using the Normal distribution.

---

### 5. **Approximate Distribution of Sample Proportions**

- The sample proportion \(\hat{p}\) of successes in \(n\) Bernoulli trials is a sum of i.i.d. Bernoulli variables.
- By CLT, \(\hat{p}\) is approximately normal for large \(n\), which is the foundation for approximate proportion tests and confidence intervals.

---

### 6. **Poisson Approximation to the Normal**

- When the Poisson parameter \(\lambda\) is large, the Poisson distribution can be approximated by a Normal distribution using the CLT, simplifying calculations.

---

### 7. **Queueing Theory and Risk Modeling**

- Aggregated random processes involving sums of random arrivals, claims, or workloads use the CLT to approximate distribution tails and quantiles.

---

### 8. **Monte Carlo Methods**

- The CLT enables **error estimation and confidence intervals** for Monte Carlo simulations, since the sample mean of simulated quantities tends to normality as the number of simulations grows.

---

### 9. **Econometrics and Finance**

- In modeling returns, aggregated risks, or residuals, the CLT justifies the use of normal distribution approximations for sums of independent shocks or errors.
- Key to many financial models assuming approximate normality of returns over longer horizons.

---

### 10. **Signal Processing and Engineering**

- In noise analysis and filtering, sums of random noise components are approximately normal by the CLT, facilitating analysis and design.

---

### Summary Table

| Application Domain         | Role of CLT                                                                                  |
|----------------------------|---------------------------------------------------------------------------------------------|
| Statistical inference       | Approximate sampling distributions, confidence intervals, tests                            |
| Proportions and counts      | Approximate normality for binomial and Poisson sums                                       |
| Simulation                  | Error bounds and distribution of estimates in Monte Carlo                                 |
| Applied probability models  | Approximations in risk, queueing, finance, signal processing                               |
| Theoretical justification  | Basis for many asymptotic results and justifying large-sample properties                   |

---

**In brief**, the Central Limit Theorem explains why the normal distribution is ubiquitous and enables approximate inference and prediction in a wide range of practical problems involving sums or averages of random variables.

---

If you want, I can provide concrete examples or explanations of CLT applications in a particular field!
User: give a list of exponential families distributions and corresponding conjugate priors
Assistant: Here is a list of common **exponential family distributions** along with their **canonical parameters** and the corresponding **conjugate prior families**:

---

| Distribution         | Support          | PDF in Canonical Form                                    | Canonical Parameter \(\eta\)                     | Conjugate Prior Family            | Comments                                |
|----------------------|------------------|---------------------------------------------------------|-------------------------------------------------|----------------------------------|----------------------------------------|
| **Bernoulli**        | \(x \in \{0,1\}\) | \( p(x|\eta) = \exp\big( x \eta - A(\eta) \big) \)       | \(\eta = \log \frac{p}{1-p}\)                    | Beta \((\alpha, \beta)\)          | Beta prior on success probability       |
| **Binomial**          | \(x = 0,1,\ldots,n\) | \(p(x|\eta) = \binom{n}{x} \exp\big(x \eta - nA(\eta)\big)\) | \(\eta = \log \frac{p}{1-p}\)                    | Beta \((\alpha, \beta)\)          | Beta prior on success probability       |
| **Poisson**           | \(x = 0,1,2,\ldots\) | \(p(x|\eta) = \exp\left( x \eta - e^\eta - \log x! \right)\) | \(\eta = \log \lambda\)                          | Gamma \((\alpha, \beta)\)         | Gamma prior on rate \(\lambda\)          |
| **Normal (known variance)** | \(x \in \mathbb{R}\) | \(p(x|\eta) = \exp\left(\eta x - \frac{\eta^2 \sigma^2}{2} - \frac{x^2}{2\sigma^2} - \frac{1}{2}\log(2\pi \sigma^2)\right)\) | \(\eta = \frac{\mu}{\sigma^2}\)                  | Normal                            | Conjugate on mean \(\mu\) with known \(\sigma^2\) |
| **Normal (both \(\mu,\sigma^2\) unknown)** | \(x \in \mathbb{R}\) | —                                                     | —                                               | Normal-Inverse-Gamma or Normal-Gamma | Prior joint on mean and variance       |
| **Exponential**       | \(x \geq 0\)     | \(p(x|\eta) = \exp(\eta x - A(\eta))\) with \(\eta = -\lambda\) | \(\eta = -\lambda\)                              | Gamma \((\alpha, \beta)\)         | Gamma prior on rate \(\lambda\)          |
| **Gamma (shape \(\alpha\) known)** | \(x \geq 0\) | \(p(x|\eta) = \exp\left(\eta \log x - \beta x - \log \Gamma(\alpha) + \alpha \log \beta \right)\) | \(\eta = \alpha - 1\) (fixed), \(\beta\) is canonical parameter | Inverse Gamma or Gamma            | Conjugate priors on scale parameter       |
| **Beta**              | \(x \in [0,1]\)  | \(p(x|\eta) = \exp\big( \eta_1 \log x + \eta_2 \log (1-x) - A(\eta) \big)\) | \(\eta = (\alpha -1, \beta -1)\)                  | None (Beta is itself prior)       | Beta is conjugate for Bernoulli/binomial  |

---

### Details and Notes

- **Exponential family form:**
\[
f(x|\theta) = h(x) \exp\left\{ \eta(\theta)^\top T(x) - A(\theta) \right\}
\]

- The **conjugate prior** has the form:
\[
p(\eta \mid \xi, \nu) \propto \exp\left\{ \eta^\top \xi - \nu A(\eta) \right\}
\]
where \(\xi\) and \(\nu\) are hyperparameters.

- **Examples:**
  - **Bernoulli/Binomial:** Beta prior on the probability parameter,
  - **Poisson:** Gamma prior on the rate,
  - **Normal (mean unknown, variance known):** Normal prior on the mean,
  - **Normal (mean and variance unknown):** Normal-Inverse-Gamma prior,
  - **Exponential:** Gamma prior on rate.

- For the **Beta distribution itself**, it serves as a conjugate prior to Bernoulli/Binomial but is also an exponential family distribution parameterized in terms of \(\alpha\), \(\beta\).

---

If you want, I can provide explicit forms of the conjugate priors, their hyperparameters, or examples of posterior updates!
User: log normal distribution
Assistant: The **Log-Normal distribution** is a continuous probability distribution of a random variable whose logarithm is normally distributed.

---

### Definition

A random variable \(X\) is **log-normally distributed** if:

\[
Y = \ln(X) \sim \mathcal{N}(\mu, \sigma^2),
\]

where \(\mu \in \mathbb{R}\) and \(\sigma > 0\).

---

### Probability Density Function (PDF)

For \(x > 0\),

\[
f_X(x; \mu, \sigma) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left( - \frac{(\ln x - \mu)^2}{2 \sigma^2} \right).
\]

Support: \( x \in (0, \infty) \).

---

### Parameters

- \(\mu\): mean of \(\ln(X)\),
- \(\sigma^2\): variance of \(\ln(X)\).

---

### Moments

- **Expectation (Mean):**
\[
\mathbb{E}[X] = e^{\mu + \frac{\sigma^2}{2}}.
\]

- **Variance:**
\[
\mathrm{Var}(X) = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2}.
\]

- **Median:**
\[
\mathrm{Median}(X) = e^{\mu}.
\]

---

### Intuition and Applications

- The log-normal distribution models quantities that are the **product of many independent positive random variables**, due to the Central Limit Theorem in the log-domain.
- Common in modeling incomes, biological measurements, sizes of particles, stock prices, and other positive-valued data with right-skewness.

---

### Summary

\[
\boxed{
\begin{cases}
f_X(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp\left( - \frac{(\ln x - \mu)^2}{2 \sigma^2} \right), \quad x > 0,\\[8pt]
\mathbb{E}[X] = e^{\mu + \frac{\sigma^2}{2}}, \\[6pt]
\mathrm{Var}(X) = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2}.
\end{cases}
}
\]

---

If you want, I can provide details on parameter estimation, related distributions, or derivations!
User: ratio of unifroms
Assistant: The **Ratio-of-Uniforms method** is a technique used for generating (sampling) random variables from a given target distribution whose density is known up to a normalizing constant. It is a type of **rejection sampling algorithm** based on transforming uniform random variables.

---

### Overview

Suppose you want to generate samples from a target density \( f(x) \) (known up to a constant), defined on the real line or a subset of \(\mathbb{R}\).

The Ratio-of-Uniforms method samples pairs \((U,V)\) from the uniform distribution over a specific region, and the ratio \( X = \frac{V}{U} \) will have the target distribution.

---

### Key Idea

1. Define the set
\[
A = \left\{ (u,v) \in \mathbb{R}^2 : u > 0, \quad 0 < u^2 \leq f\left(\frac{v}{u}\right) \right\}.
\]

2. Sample \((U,V)\) uniformly from \(A\).

3. Then
\[
X = \frac{V}{U}
\]
has density proportional to \(f(x)\).

---

### Algorithm Steps

- Find a bounding rectangle that contains the set \(A\), say \( (u,v) \in [0, u_{\max}] \times [v_{\min}, v_{\max}] \).
- Repeat:
  - Generate \(U \sim \text{Uniform}(0, u_{\max})\),
  - Generate \(V \sim \text{Uniform}(v_{\min}, v_{\max})\),
  - If \((U,V) \in A\), accept \(X = V/U.\)
- Otherwise, reject and resample.

---

### Advantages

- Works well for unimodal distributions,
- Can be efficient compared to traditional rejection samplers when \(f(x)\) is complicated,
- Does **not** require the normalizing constant of \(f\),
- Can be used for densities on unbounded domains.

---

### Conditions & Variants

- The method requires the set \(A\) to be bounded to define the bounding rectangle,
- Variants exist to handle multidimensional cases and improve efficiency,
- The method is sometimes generalized via transformations.

---

### Intuition

By sampling uniformly from the 2D region defined by \(u^2 \leq f(v/u)\), the marginal distribution of \(V/U\) matches the target density.

---

### References

- Kinderman, A.J. and Monahan, J.F. (1977). “Computer generation of random variables using the ratio of uniform deviates.” ACM Transactions on Mathematical Software.
- Robert, C.P. and Casella, G. (2004). *Monte Carlo Statistical Methods*. Springer.

---

If you want, I can provide a **concrete example** or **pseudocode** for the Ratio-of-Uniforms method.
User: asymptotic variance of MLE 
Assistant: The **asymptotic variance of the Maximum Likelihood Estimator (MLE)** is a key result in classical statistics describing the distribution of the MLE as the sample size \( n \to \infty \).

---

### Setup

- Let \( X_1, X_2, \ldots, X_n \) be i.i.d. with density \( f(x; \theta) \), where \(\theta \in \Theta \subseteq \mathbb{R}^d\) is the parameter.
- Let \(\hat{\theta}_n\) be the MLE based on the sample.

---

### Asymptotic Distribution of MLE

Under regularity conditions, the MLE \(\hat{\theta}_n\) is **asymptotically normal:**

\[
\sqrt{n}(\hat{\theta}_n - \theta_0) \xrightarrow{d} \mathcal{N}\left(0, I(\theta_0)^{-1}\right)
\]

where:

- \(\theta_0\) is the true parameter,
- \(I(\theta_0)\) is the **Fisher Information matrix** at \(\theta_0\),
- \(I(\theta_0)^{-1}\) is the **asymptotic covariance matrix** of \(\hat{\theta}_n\).

---

### Definition of Fisher Information Matrix

\[
I(\theta) = \mathbb{E}_\theta \left[ \left(\frac{\partial}{\partial \theta} \log f(X; \theta)\right) \left(\frac{\partial}{\partial \theta} \log f(X; \theta)\right)^\top \right]
\]

Equivalently, under conditions:

\[
I(\theta) = - \mathbb{E}_\theta \left[ \frac{\partial^2}{\partial \theta \partial \theta^\top} \log f(X; \theta) \right].
\]

---

### Asymptotic Variance

- The variance-covariance matrix of \(\hat{\theta}_n\) shrinks at rate \(1/n\),
- The **asymptotic variance** of \(\hat{\theta}_n\) is:

\[
\mathrm{Var}(\hat{\theta}_n) \approx \frac{1}{n} I(\theta_0)^{-1}.
\]

For a scalar parameter \(\theta\), this reduces to:

\[
\mathrm{Var}(\hat{\theta}_n) \approx \frac{1}{n I(\theta_0)}.
\]

---

### Intuition

- The MLE behaves like the average of independent score contributions,
- The Fisher Information measures the amount of information per observation,
- As \(n\) grows, the estimator becomes approximately normal with variance inversely proportional to \(n\).

---

### Summary:

\[
\boxed{
\hat{\theta}_n \approx \mathcal{N}\left( \theta_0, \frac{1}{n} I(\theta_0)^{-1} \right),
\quad\text{and}\quad
\mathrm{Var}(\hat{\theta}_n) \approx \frac{1}{n} I(\theta_0)^{-1}.
}
\]

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If you want, I can provide:

- A derivation sketch,
- Explicit examples for specific models (e.g., Normal),
- How to estimate \(I(\theta)\) from data.
User: binomial coefficient
Assistant: