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I want you to teach me Bayesian Statistics from 0 to 100. Just start with 1st lecture of 100lectures. Go on!

Let's go to the next lecture till we make it to 100th lectures!

Before moving on to the next lesson, explain more about MCMC, including basic formulas, with specific examples. Create a problem for me to solve and show me the answer.

I don't think I can implement a Markov chain myself with this explanation, please be more specific with formulas and examples so that I can implement and use a Markov chain myself.

Let's go back to the description of MCMC. What is a Stationary Distribution?

What exactly does it mean for a probability distribution to be stationary in a Markov chain? If I have a set of states and transition matrix with values [[0.9, 0.1], [0.2, 0.8]] , is [0.9, 0.1] the probability distribution?

How can a vector * Matrix product be understood intuitively?

A Jacobian matrix is a local linear transformation matrix, so what does a Hessian matrix mean geometrically or algebraically, and how does it have that meaning?

If the i-row j-column elements of a Jacobian matrix are \frac{\partial x_j}{\partial y_i} where y_i is the ith element of vector y of length n, and x_j is the jth element of vector x of length n. Then what are the elements of column i, row j of the Hessian matrix?

If a function whose output has dimension d(>1) is called a multivariate function, what should we call a function whose input has dimension d(>1)?

What if it's not a unit vector?

v가 단위벡터면 어떻게 해서 그 값이 rate를 여전히 알려주는거야?

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Add code to resize each picture while maintaining the original proportions.

Change the size of the two images to be fixed.

When we apply the Hessian matrix to another function, what we are actually doing is transforming the second function in a way that is informed by the behavior of the first function. In other words, the Hessian matrix of the first function provides information about the local curvature of the function, and this information is used to transform the second function in a way that reflects that curvature.

Here is a simple example to illustrate the idea. Let's say that we have two functions, f(x, y) and g(x, y). We can think of the Hessian matrix of f(x, y) as a linear transformation, H, that operates on g(x, y) to produce a new function h(x, y). The operation can be written as h(x, y) = H(g(x, y)).

In matrix form, we can write the operation as follows:

$\begin{bmatrix} \frac{\partial^2f}{\partial x^2} & \frac{\partial^2f}{\partial x\partial y} \ \frac{\partial^2f}{\partial y\partial x} & \frac{\partial^2f}{\partial y^2} \end{bmatrix} \begin{bmatrix} \frac{\partial g}{\partial x} \ \frac{\partial g}{\partial y} \end{bmatrix} = \begin{bmatrix} \frac{\partial h}{\partial x} \ \frac{\partial h}{\partial y} \end{bmatrix}$

In this example, the Hessian matrix of f(x, y) acts on the gradient of g(x, y) to produce the gradient of the new function h(x, y). The new function h(x, y) will have properties that reflect the curvature of f(x, y), and these properties can be useful in various optimization and machine learning algorithms.

H=[
∂x
2

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f
​

∂y∂x
∂
2
f
​

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∂x∂y
∂
2
f
​

∂y
2

∂
2
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​

​
]