## The maximum a posteriori probability of Bayesian inference (MAP)

This paper records the mathematical principle of Bayesian posterior probability distribution in detail , A binary classification problem based on Bayesian posterior probability , Let's talk about my understanding of Bayesian inference .

### 1. Dichotomous problem

Given N The data set of 2 samples , use \(X\) To express , Each sample \(x_n\) There are two properties , Finally, it belongs to a certain category \(t\)

\(t=\left\{0,1\right\}\)

\(\mathbf{x_n}=\begin{pmatrix}x_{n1} \\ x_{n2} \\ \end{pmatrix}\), Suppose the model parameters \(w=\begin{pmatrix} w_1 \\ w_2\end{pmatrix}\)

\(\mathbf{X}=\begin{bmatrix} x_1^T \\ x_2^T \\. \\. \\ x_n^T\end{bmatrix}\)

Picture the sample set as follows ：

According to Bayesian formula, there are ：

\[p(w|t,X)=\frac {p(t|X,w)p(w)} {p(t|X)} \] ( The formula 1)

\(p(w | t,X)\) Tell us ： In a given training sample set \(X\) And a classification of these samples \(t\) ( It's a supervised learning , Because we already have a sample set \(X\)、 And the classification of each sample in the sample set \(t\)), You need to solve the model parameters \(w\) . therefore ,\(w\) It is unknown. , It needs to be solved by Bayesian probability formula according to samples . We have found \(p(w|t,X)\) The distribution of , So you know the model parameters \(w\)

When we get the optimal model parameters \(w^*\) after , Given a sample to be predicted \(\mathbf{x_{new}}\) According to the formula \[P(T_{new}=1|x_{new}, w^*)\] To calculate the new sample \(\mathbf{x_{new}}\) Classified as 1 What's the probability of , That's what the model predicts .

The formula 1 The medium size has three parts on the right ,\(p(t|X,w)\) It's called likelihood probability (likelihood),\(p(w)\) It's called a priori probability , These two parts are generally easier to solve . The hardest part to solve is the denominator : \(p(t|X)\) It's called the boundary likelihood function (marginal likelihood). But fortunately , Boundary likelihood function and model parameters \(w\) irrelevant , therefore , You can think of the denominator as about \(w\) A constant of .

In Mathematics , If the prior probability is conjugate with the likelihood probability , So the posterior distribution probability \(p(w|t, X)\) It follows the same distribution as the prior probability . for instance ： A priori probability obeys Beta Distribution , Likelihood probability follows binomial distribution , The prior probability distribution is conjugate with the likelihood probability distribution , Then the posterior probability also obeys Beta Distribution .

therefore , When using Bayes formula , If the selected prior probability distribution is conjugate with the likelihood probability distribution , The posterior probability distribution can be easily calculated （ Or to be able to accurately calculate a ** Concrete / The precise ** Model parameters \(w^*\)）, This is it. ：can compute posterior **analytically**. But the reality is , They are not conjugate , So there are three common approximate calculation methods ：

- Point estimation (Point Estimate--MAP Method )
- Laplace approximation method (Laplace approximation)
- Sampling method (Sampling--Metropolis-Hastings)

This paper only introduces the point estimation method .

Back to the formula 1, Let's start with a priori probability \(p(w)\) , A priori probability is similar to when making a decision , Existing experience . Because we already have training samples \(X\), Draw the contour lines corresponding to these samples , They are very close to Gaussian distribution , Then it can be considered that the prior probability obeys Gaussian distribution . That is to say \[p(w)=N(0,\sigma^2I)\]. among ,\(w\) It's a vector ,\(I\) It's a unit matrix .

And then there's likelihood probability \(p(t|X,w)\) , Suppose that given the model parameters \(w\) And sample sets \(X\) Under the condition of , The classification results of each sample are independent of each other , therefore ：

\[p(t|X,w)=\prod_{n=1}^N p(t_n|x_n, w)\] ( The formula 2)

for instance , When the model parameters are known \(w\) when ,\(w\) take \(x_1\) The prediction is positive , take \(x_2\) The prediction is negative …… take \(x_n\) Predict the positive , The prediction results of each sample are independent of each other . namely ：\(w\) Yes \(x_1\) Forecast results of It won't affect its effect on \(x_2\) Forecast results of .

because , It's a dichotomy problem ,\(t_n=\left\{0,1\right\}\) , We can further extend the formula 2 It's written in ：\[p(t|X,w)=\prod_{n=1}^N p(T_n=t_n|x_n, w)\] , among \(T_n\) Representative sample \(x_n\) Classified into a certain class A random variable ,\(t_n\) Is the value of the random variable . such as \(t_n=0\) Presentation sample \(x_n\) Be classified as positive ,\(t_n=1\) Means to be classified as negative .

### 2. sigmod function

Because the probability of a random variable taking a certain value is in [0,1] Between , So ask for the solution \(p(t|X,w)\), Our goal is ： Find a function \(f(\mathbf{x_n};w)\) This function produces a probability value . To simplify the discussion , choice \(sigmod(w^T*x)\) , therefore ：

\[P(T_n=1|x_n,w)=\frac{1}{1+exp(-w^T*x_n)}\]

that ：

\[P(T_n=0|x_n,w)=1-P(T_n=1|x_n,w)=\frac{exp(-w^T*x_n)}{1+exp(-w^T*x_n)}\]

Combine the above two formulas into one ：

\[P(T_n=t_n|x_n,w)=P(T_n=1|x_n,w)^{t_n}P(T_n=0|x_n,w)^{1-t_n}\]

about N Samples , The formula 2 Can be written as ：

\[p(t|X,w)=\prod_{n=1}^N (\frac{1}{1+exp(-w^T*x_n)})^{t_n}(\frac{exp(-w^T*x_n)}{1+exp(-w^T*x_n)})^{1-t_n}\]

( The formula 3)

thus , The prior probability follows Gaussian distribution , The likelihood probability is determined by the formula 3 give , Then we can solve the posterior probability formula mentioned above ：\[p(w|X,t,\sigma^2)\]

As long as we get the posterior probability , You can use the following formula to calculate the probability that the new sample will be divided into negative categories ：

\[P(t_{new}=1|x_{new}, X, t)=E_{p(w|X,t,\sigma^2)}\left(\frac{1}{1+exp(-w^T*x_{new})}\right)\]

Explain the formula ： Because we've got the posterior probability \(p(w|X,t,\sigma^2)\) The expression of , It's about \(f(x_n;w)\) Function of , Calculate the expected value of this function \(E\), The expectation is Predict new samples \(x_{new}=1\) Probability .

good , The next step is to solve the posterior probability .

### 3. Find the posterior probability

I've said that before , The prior probability follows Gaussian distribution \(N(0,\sigma^2I)\), The likelihood distribution is determined by ** The formula 3** give , And the denominator -- The boundary likelihood function is a function of \(w\) The constant , So define a function \(g(w;X,t,\sigma^2)=p(t|X,w)p(w|\sigma^2)\) , function \(g\) Obviously with posterior probability \(p(w|X,t,\sigma^2)\) Is proportional to the . therefore , We get the function \(g\) The maximum of , It is equivalent to finding the optimal parameter of posterior probability \(w^*\).

The problem here is ： How can we maximize the function g Well ？\(g\) yes \(w\) Function of ,\(w\) Which value does the function take \(g\) To the maximum ？

Here we need to use a method called Newton's method (Newton-Raphson method). Newton's method can be used to ** Looking for zeros in a function **. It works through the following formula ：

\[x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\]

Continuous iteration , Finally, we find that the value of the function is 0 The point of .

In mathematics, when the judgment function takes the extreme value at a certain point , There are the following theorems ：

Let's take the derivative function of one variable \(h(x)\) For example ,\(h(x)\) Derivative is 0 The point of is the extreme point , But this extreme point is a minimum , Or the maximum value ？ In this case, we can judge \(h(x)\) It is the second derivative to judge whether the extreme point is a minimum or a maximum . if \(h'(x_n)=0\) And \(h''(x_n)<0\), be , be \(h(x)\) In \(x_n\) Take the maximum .

therefore , If you can judge \(g(w;X,t,\sigma^2)\) About \(w\) The second derivative of is less than 0, Then you can use Newton's method to solve \(g(w;X,t,\sigma^2)\) The first derivative of is about \(w\) Zero point of , namely \(g'(w;X,t,\sigma^2)=0\) when \(w\) The values for \(w_0\), This \(w_0\) It's the best solution \(w^*\) 了 .

good , Let's prove it \(g(w;X,t,\sigma^2)\) About \(w\) The second derivative of is less than 0 Of . because \(w\) It's a vector , In multivariate functions , It's equivalent to proving that ：\(g(w;X,t,\sigma^2)\) About \(w\) The Hessian matrix of is negative definite .

Will function \(g\) Take the logarithm , Maximize \(log (g(w;X,t,\sigma^2))\)

\(log (g(w;X,t,\sigma^2))=log({p(t|X,w)p(w|\sigma^2}))\)

\[=log(p(t|X,w)+log(p(w|\sigma^2)\]

To simplify the formula , Make the following agreement ：

hypothesis \(w\) It's a \(D\) Dimension vector ：

The first three terms are after the prior distribution obeys the Gaussian distribution , The result of simplification . According to the vector derivative formula ：\[\frac{\partial w^Tw}{\partial w}=w\]

By the chain derivation rule ：

obtain ：

therefore ：\(log (g(w;X,t,\sigma^2))\) Yes \(w\) The first partial number of is as follows ：

The second partial derivative is as follows ：

\(I\) It's a unit matrix ,\(0=<P_n<=1\) It's a probability value , The second partial derivative is the Hessian matrix , It's negative .

The proof is complete .

** thus , You can safely use Newton's method to iterate , find \(g(w;X,t,\sigma^2)\) When the maximum value is taken, the parameter \(w\) The value of the , And this value is \(w^*\)**

Now? ,\(w^*\) Come on , You can use the following formula to predict new samples \(x_{new}\) Predicted to be negative (\(T_{new}\) The value is 1) The probability of

\[P(T_{new}=1|x_{new},w^*)=\frac{1}{1+exp(-w^{*T}x_{new})}\]

### decision boundary

Because it's a binary problem , Let's look at classification using Bayesian posterior probability Decide the boundary What does it look like . Because the output is a probability value , It's obvious that \(P(T_{new}=1|x_{new},w^*)>0.5\) The prediction is negative ,\(P(T_{new}=1|x_{new},w^*)<0.5\) The prediction is positive . That's equal to 0.5 When ？

according to ：\[P(T_{new}=1|x_{new},w^*)=\frac{1}{1+exp(-w^{*T}x_{new})}=0.5\] obtain ：

\[-w^{*T}*x=0=w_1^*x_1+w_2^*x_2\]

\[x_2=\frac{w_1^*}{w_2^*}*(-x_1)\]

That is to say, samples \(\mathbf{x}=\begin{pmatrix}x_{1} \\ x_{2} \\ \end{pmatrix}\) Two properties of \(x_1\) and \(x_2\) It's linearly proportional . And this straight line is decision boundary

### summary

Bayesian method is a common method in machine learning , There are three parts in Bayesian formula , A priori probability distribution function 、 Likelihood probability distribution function 、 And the boundary likelihood probability distribution function （ The denominator of Bayesian formula ）. The three parts are worked out , Then we get the posterior probability distribution , And then for a new sample \(x_{new}\) Calculate the expected value of the posterior probability distribution , This expectation is the prediction result of Bayesian model .

Because the calculation of posterior probability distribution depends on prior probability distribution function 、 Likelihood probability distribution function , When the two are conjugate , Posterior probability and prior probability obey the same distribution function , So we can deduce and calculate the posterior probability distribution (posterior could be computed analytically). however , When the two are not conjugate , It is an approximation of the posterior probability distribution . There are three ways to calculate approximations , Point estimation (point estimate --- MAP), Laplace approximation ,Metropolis-Hastings Sampling method . And this article mainly introduces It's the first way ： Point estimation (point estimate --- maximum a posteriori).

maximum a posteriori Where is the maximization of the quality of life ？ In fact, it is reflected in the maximization of the likelihood distribution function . The negative definiteness of Hesse matrix proves \(g(w;X,t,\sigma^2)\) With maximum , And then we use Newton's method to find this function \(g\) Take the optimal parameter solution of the maximum value \(w^*\). And the optimal parameters are obtained \(w^*\), The posterior probability distribution formula is obtained . For a new sample to be predicted \(x_{new}\), Calculate the expected value of the posterior probability distribution of the sample , The expected value is the prediction result of Bayesian model for new samples .

### Reference material

Newton method ：https://zh.wikipedia.org/wiki/ Newton method

Blog Garden Markdown The formula is messy ：http://www.cnblogs.com/cmt/p/markdown-latex.html

original text ：http://www.cnblogs.com/hapjin/p/8834794.html

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