Skip to main content

My Bayes algorithm, for the name of Thomas Bayes.

Project description

# Bayes Classifier

## Principle

### Naive Bayes

p(c|x)=\frac{p(x|c)p(c)}{p(x)}\sim p(x|c)p(c)\\
\sim \prod_ip(x_i|c)p(c) = \prod_ip(x_i,c)p(c)^{1-n}~~~~~~~~~\text{(Naive condition)}\\

### Semi Naive Bayes

p(c|x,y)=\sim p(x|c)p(c|y)\\
\sim \prod_ip(x_i|c)p(c|y) ~~~~~~~~~\text{(Semi-Naive condition)}

where $p(c|y)​$ will be estimated by say of neural networks.

### Hemi Naive Bayes, in more general form

When $y$ is empty, it is equiv. to the naive one.

p(c|x,y_1,\cdots y_m)
\sim \prod_ip(x_i|c)\prod_ip(c|y_i)p(c)^{1-m} ~~~~~~~~~~(Hemi-condition)\\
\sim \prod_ip(x_i|c)\prod_if_c(y_i)p(c)^{1-m}\\
\sim \prod_ip(x_i,c)\prod_if_c(y_i)p(c)^{1-m-n}

## Predict

\frac{p(c|x,y)}{p(c'|x,y)}= \prod_i(\frac{p(x_i|c)}{p(x_i|c')})\frac{p(c|y)}{p(c'|y)}\\
= \prod_i(\frac{p(x_i,c)}{p(x_i,c')})\frac{p(c|y)}{p(c'|y)}(\frac{p(c')}{p(c)})^n
~~~~~~~~~\text{(Semi-Naive condition)}\\
\sim \prod_i(\frac{N(x_i,c)}{N(x_i,c')})\frac{p(c|y)}{p(c'|y)}(\frac{N(c')}{N(c)})^n ~~~~~~~~~~~~~~~~~~~~~\text{(estimate)}

\frac{p(c|x,y_1,\cdots, y_m)}{p(c'|x,y_1,...,y_m)}\sim ... (\frac{N(c')}{N(c)})^{n+m-1}\prod_i\frac{p(c|y_i)}{p(c'|y_i)} ~~~~~~~~~(\text{Hemi-condition})

### 0-1 cases

r = \frac{p(1|x,y)}{p(0|x,y)}\sim \prod_i(\frac{N(x_i,1)}{N(x_i,0)})\frac{p(1|y)}{1-p(1|y)}(\frac{N(0)}{N(1)})^n (Semi)\\

r \sim \prod_i(\frac{N(x_i,1)}{N(x_i,0)})\prod_i\frac{p(1|y_i)}{1-p(1|y_i)}(\frac{N(0)}{N(1)})^{n+m-1} (Hemi)

iff $r\geq 1$, $(x,y)$ is in class 1, else in class 0.

## Estimate (for continuous rv)

$p(x)\sim \frac{N(x)}{N}, N(x):$ the number of samples in a neighborhood of $x$

Project details

Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

Files for tomas, version 0.1.1
Filename, size File type Python version Upload date Hashes
Filename, size tomas-0.1.1.tar.gz (2.2 kB) File type Source Python version None Upload date Hashes View

Supported by

Pingdom Pingdom Monitoring Google Google Object Storage and Download Analytics Sentry Sentry Error logging AWS AWS Cloud computing DataDog DataDog Monitoring Fastly Fastly CDN DigiCert DigiCert EV certificate StatusPage StatusPage Status page