Pattern Recognition - Chapter 3: Bayes decision rule
![](https://static.wixstatic.com/media/64c52b_74d426446194451eacc4eddccae0454b~mv2.png/v1/fill/w_320,h_244,al_c,q_85,enc_auto/64c52b_74d426446194451eacc4eddccae0454b~mv2.png)
(*Source: Richard O.Duda et al. Pattern Recognition)
1. Bayes decision rule
1.1. Risk function
![](https://static.wixstatic.com/media/64c52b_fc04aaa61b4845cf8078040151affe0c~mv2.png/v1/fill/w_314,h_105,al_c,q_85,enc_auto/64c52b_fc04aaa61b4845cf8078040151affe0c~mv2.png)
1.1.1. Simple cost function
![](https://static.wixstatic.com/media/64c52b_6b14bde7545048b7ac870cb02b857949~mv2.jpg/v1/fill/w_288,h_320,al_c,q_80,enc_auto/64c52b_6b14bde7545048b7ac870cb02b857949~mv2.jpg)
(*Source: http://www.writeups.org/songoku-age-23-dragon-ball/)
![](https://static.wixstatic.com/media/64c52b_b0ba5a779f604b6b9f7ca263ba9bc890~mv2.png/v1/fill/w_336,h_47,al_c,q_85,enc_auto/64c52b_b0ba5a779f604b6b9f7ca263ba9bc890~mv2.png)
Equation (1)
1.1.2. Risk function depends the correct class
![](https://static.wixstatic.com/media/64c52b_e55910280f55429babb1125e4954fbc4~mv2.jpg/v1/fill/w_200,h_259,al_c,q_80,enc_auto/64c52b_e55910280f55429babb1125e4954fbc4~mv2.jpg)
(*Source: http://dragonballwiki.de/Son-Goku)
![](https://static.wixstatic.com/media/64c52b_6af98748c76945f6ba56e8608a8a0811~mv2.png/v1/fill/w_312,h_100,al_c,q_85,enc_auto/64c52b_6af98748c76945f6ba56e8608a8a0811~mv2.png)
![](https://static.wixstatic.com/media/64c52b_045ebafff34b4d17965944c282ded4e0~mv2.png/v1/fill/w_299,h_54,al_c,q_85,enc_auto/64c52b_045ebafff34b4d17965944c282ded4e0~mv2.png)
Equation (2)
1.1.3. Risk function over all samples
![](https://static.wixstatic.com/media/64c52b_4045183d0be644aa91988268af2734a6~mv2.jpg/v1/fill/w_311,h_320,al_c,q_80,enc_auto/64c52b_4045183d0be644aa91988268af2734a6~mv2.jpg)
(*Source:www.xemgame.com/ngoc-rong-dai-chien-khi-gioi-cosplay-dau-dau-voi-kieu-toc-cua-songoku-post144659.html)
![](https://static.wixstatic.com/media/64c52b_fab2cbc4177e4eb48842bdc171b576c0~mv2.png/v1/fill/w_310,h_96,al_c,q_85,enc_auto/64c52b_fab2cbc4177e4eb48842bdc171b576c0~mv2.png)
![](https://static.wixstatic.com/media/64c52b_4528d0850bec448dbfd7a515f973cc88~mv2.png/v1/fill/w_289,h_60,al_c,q_85,enc_auto/64c52b_4528d0850bec448dbfd7a515f973cc88~mv2.png)
![](https://static.wixstatic.com/media/64c52b_60ed50f12b7c439485c9d7ff66ac9fac~mv2.png/v1/fill/w_404,h_64,al_c,q_85,enc_auto/64c52b_60ed50f12b7c439485c9d7ff66ac9fac~mv2.png)
Equation (3)
1.2. Bayes decision rule
![](https://static.wixstatic.com/media/64c52b_12e30800611e443b948092873d8586aa~mv2.png/v1/fill/w_320,h_52,al_c,q_85,enc_auto/64c52b_12e30800611e443b948092873d8586aa~mv2.png)
![](https://static.wixstatic.com/media/64c52b_d5c5aaedd43c4dc29d02abfa331de3a9~mv2.png/v1/fill/w_634,h_64,al_c,q_85,enc_auto/64c52b_d5c5aaedd43c4dc29d02abfa331de3a9~mv2.png)
![](https://static.wixstatic.com/media/64c52b_e2315c329aa94ca487c9049e4fe4f598~mv2.png/v1/fill/w_583,h_47,al_c,q_85,enc_auto/64c52b_e2315c329aa94ca487c9049e4fe4f598~mv2.png)
![](https://static.wixstatic.com/media/64c52b_387ca77c7e3d4a1ca9aa25fcc25ce538~mv2.png/v1/fill/w_237,h_56,al_c,q_85,enc_auto/64c52b_387ca77c7e3d4a1ca9aa25fcc25ce538~mv2.png)
![](https://static.wixstatic.com/media/64c52b_0ec3505237bd42838782de6ff0da5895~mv2.png/v1/fill/w_233,h_36,al_c,q_85,enc_auto/64c52b_0ec3505237bd42838782de6ff0da5895~mv2.png)
![](https://static.wixstatic.com/media/64c52b_6db383e2c10744fdaf52eaef86db6ba9~mv2.png/v1/fill/w_201,h_39,al_c,q_85,enc_auto/64c52b_6db383e2c10744fdaf52eaef86db6ba9~mv2.png)
Equation (4)
2. Discriminant functions and decision boundaries
2.1. Discriminant function
![](https://static.wixstatic.com/media/64c52b_9db2585a4aec410a875953e8f71f0859~mv2.png/v1/fill/w_320,h_168,al_c,q_85,enc_auto/64c52b_9db2585a4aec410a875953e8f71f0859~mv2.png)
(*Source: Richard O.Duda et al. Pattern Recognition)
![](https://static.wixstatic.com/media/64c52b_cf37138b56584d549050c89adc250b59~mv2.png/v1/fill/w_199,h_44,al_c,q_85,enc_auto/64c52b_cf37138b56584d549050c89adc250b59~mv2.png)
![](https://static.wixstatic.com/media/64c52b_44e8677023d541dab058542e09ea9acc~mv2.png/v1/fill/w_259,h_40,al_c,q_85,enc_auto/64c52b_44e8677023d541dab058542e09ea9acc~mv2.png)
![](https://static.wixstatic.com/media/64c52b_f645794211f04e34a6bfa6605e27f8fb~mv2.png/v1/fill/w_320,h_35,al_c,q_85,enc_auto/64c52b_f645794211f04e34a6bfa6605e27f8fb~mv2.png)
2.2. Decision boundary
![](https://static.wixstatic.com/media/64c52b_5ce4c1ad68d84c1fa9adc8ec5e9d9ab8~mv2.png/v1/fill/w_320,h_307,al_c,q_85,enc_auto/64c52b_5ce4c1ad68d84c1fa9adc8ec5e9d9ab8~mv2.png)
(*Source: Richard O.Duda et al. Pattern Recognition)