Pattern Recognition - Chapter 5: Maximum-Likelihood estimation
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1. Maximum-Likelihood estimate (MLE)
![](https://static.wixstatic.com/media/64c52b_36849d53fe6c4036b0d1489a201b6e7d~mv2.png/v1/fill/w_343,h_26,al_c,q_85,enc_auto/64c52b_36849d53fe6c4036b0d1489a201b6e7d~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b44e7d9cf2c84bb1ba1e8a058e62de81~mv2.png/v1/fill/w_122,h_42,al_c,q_85,enc_auto/64c52b_b44e7d9cf2c84bb1ba1e8a058e62de81~mv2.png)
Equation (1)
![](https://static.wixstatic.com/media/64c52b_bbdc7ffa127b476fb082d1d24621f7f5~mv2.png/v1/fill/w_227,h_53,al_c,q_85,enc_auto/64c52b_bbdc7ffa127b476fb082d1d24621f7f5~mv2.png)
![](https://static.wixstatic.com/media/64c52b_10358f0e7717439dae70543ec3eb23ec~mv2.png/v1/fill/w_320,h_309,al_c,q_85,enc_auto/64c52b_10358f0e7717439dae70543ec3eb23ec~mv2.png)
(*Source: Richard O.Duda et al. Pattern Recognition)
![](https://static.wixstatic.com/media/64c52b_b2478b430fb447b08c28ed4ffc712556~mv2.png/v1/fill/w_109,h_30,al_c,q_85,enc_auto/64c52b_b2478b430fb447b08c28ed4ffc712556~mv2.png)
![](https://static.wixstatic.com/media/64c52b_09609ddd701f4ae4886ba48e499ffbfa~mv2.png/v1/fill/w_144,h_66,al_c,q_85,enc_auto/64c52b_09609ddd701f4ae4886ba48e499ffbfa~mv2.png)
![](https://static.wixstatic.com/media/64c52b_a9de07921b95410e81d97deb0fe9dd8f~mv2.png/v1/fill/w_490,h_56,al_c,q_85,enc_auto/64c52b_a9de07921b95410e81d97deb0fe9dd8f~mv2.png)
Equation (2)
![](https://static.wixstatic.com/media/64c52b_94f9704118a347dca4ecb36cbaee4d6d~mv2.png/v1/fill/w_307,h_80,al_c,q_85,enc_auto/64c52b_94f9704118a347dca4ecb36cbaee4d6d~mv2.png)
![](https://static.wixstatic.com/media/64c52b_cc7537cf16064a4fbff3d5223e9bf28a~mv2.png/v1/fill/w_333,h_78,al_c,q_85,enc_auto/64c52b_cc7537cf16064a4fbff3d5223e9bf28a~mv2.png)
![](https://static.wixstatic.com/media/64c52b_4e7303360dda4df3b85d77d7494d4792~mv2.png/v1/fill/w_162,h_67,al_c,q_85,enc_auto/64c52b_4e7303360dda4df3b85d77d7494d4792~mv2.png)
Equation (3)
2. MLE in Gaussian distribution (unknown mean)
![](https://static.wixstatic.com/media/64c52b_319e191ea38441859e06e8266462b7cb~mv2.png/v1/fill/w_160,h_64,al_c,q_85,enc_auto/64c52b_319e191ea38441859e06e8266462b7cb~mv2.png)
![](https://static.wixstatic.com/media/64c52b_09060a37f68f454eb0695f1d2640935b~mv2.png/v1/fill/w_369,h_73,al_c,q_85,enc_auto/64c52b_09060a37f68f454eb0695f1d2640935b~mv2.png)
![](https://static.wixstatic.com/media/64c52b_3addb09c61c84d18a8dd7fb6322faf27~mv2.png/v1/fill/w_431,h_62,al_c,q_85,enc_auto/64c52b_3addb09c61c84d18a8dd7fb6322faf27~mv2.png)
![](https://static.wixstatic.com/media/64c52b_9518c0beb3cf4da08594ac9a0d4bb331~mv2.png/v1/fill/w_484,h_68,al_c,q_85,enc_auto/64c52b_9518c0beb3cf4da08594ac9a0d4bb331~mv2.png)
![](https://static.wixstatic.com/media/64c52b_e9a56c745c294036a19d7d52edfdf9b8~mv2.png/v1/fill/w_281,h_64,al_c,q_85,enc_auto/64c52b_e9a56c745c294036a19d7d52edfdf9b8~mv2.png)
Equation (4)
![](https://static.wixstatic.com/media/64c52b_3c6b7b076df94847936b1e93927b52dd~mv2.png/v1/fill/w_139,h_33,al_c,q_85,enc_auto/64c52b_3c6b7b076df94847936b1e93927b52dd~mv2.png)
Equation (5)
![](https://static.wixstatic.com/media/64c52b_e8ddc12ddd9647659acf67f54b8c6e6e~mv2.png/v1/fill/w_462,h_57,al_c,q_85,enc_auto/64c52b_e8ddc12ddd9647659acf67f54b8c6e6e~mv2.png)
Equation (6)
![](https://static.wixstatic.com/media/64c52b_5aa3d5b9e579477cae7d6d071e93ee95~mv2.png/v1/fill/w_165,h_59,al_c,q_85,enc_auto/64c52b_5aa3d5b9e579477cae7d6d071e93ee95~mv2.png)
![](https://static.wixstatic.com/media/64c52b_d236213f55a24e45b10eac137df367a3~mv2.png/v1/fill/w_128,h_64,al_c,q_85,enc_auto/64c52b_d236213f55a24e45b10eac137df367a3~mv2.png)
![](https://static.wixstatic.com/media/64c52b_099b41781f6443fd9cff27f91296921b~mv2.png/v1/fill/w_147,h_65,al_c,q_85,enc_auto/64c52b_099b41781f6443fd9cff27f91296921b~mv2.png)
![](https://static.wixstatic.com/media/64c52b_71dee28a49124ad88b5b3e6579a646cd~mv2.png/v1/fill/w_156,h_78,al_c,q_85,enc_auto/64c52b_71dee28a49124ad88b5b3e6579a646cd~mv2.png)
![](https://static.wixstatic.com/media/64c52b_060058a8e204465f8255aef11c58f666~mv2.png/v1/fill/w_127,h_60,al_c,q_85,enc_auto/64c52b_060058a8e204465f8255aef11c58f666~mv2.png)
![](https://static.wixstatic.com/media/64c52b_cd3fc6d391b541ca8babcdc1b06be245~mv2.png/v1/fill/w_93,h_66,al_c,q_85,enc_auto/64c52b_cd3fc6d391b541ca8babcdc1b06be245~mv2.png)
3. MLE in Gaussian distribution (unknown mean and covariance)
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![](https://static.wixstatic.com/media/64c52b_0b4980128e8342dda550dcb8d3ed32ad~mv2.png/v1/fill/w_297,h_66,al_c,q_85,enc_auto/64c52b_0b4980128e8342dda550dcb8d3ed32ad~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ec7ef90586414ec3971dbbed2724cff5~mv2.png/v1/fill/w_224,h_89,al_c,q_85,enc_auto/64c52b_ec7ef90586414ec3971dbbed2724cff5~mv2.png)
![](https://static.wixstatic.com/media/64c52b_5f8b60d1172243be823c62a83fb349b6~mv2.png/v1/fill/w_95,h_65,al_c,q_85,enc_auto/64c52b_5f8b60d1172243be823c62a83fb349b6~mv2.png)
![](https://static.wixstatic.com/media/64c52b_ff78cfd07d1b43ee875a72beffdab28d~mv2.png/v1/fill/w_154,h_62,al_c,q_85,enc_auto/64c52b_ff78cfd07d1b43ee875a72beffdab28d~mv2.png)
![](https://static.wixstatic.com/media/64c52b_8319432ac33e492cb51eb718476fe10c~mv2.png/v1/fill/w_95,h_65,al_c,q_85,enc_auto/64c52b_8319432ac33e492cb51eb718476fe10c~mv2.png)
![](https://static.wixstatic.com/media/64c52b_aa830d92a912408e97484e4bc7ad6d27~mv2.png/v1/fill/w_201,h_71,al_c,q_85,enc_auto/64c52b_aa830d92a912408e97484e4bc7ad6d27~mv2.png)
4. MLE in Poisson distribution
![](https://static.wixstatic.com/media/64c52b_d8f20afbedd84eafa52317b2d8d2612f~mv2.png/v1/fill/w_318,h_70,al_c,q_85,enc_auto/64c52b_d8f20afbedd84eafa52317b2d8d2612f~mv2.png)
![](https://static.wixstatic.com/media/64c52b_1a9016f54c8e4035af555cf82356d907~mv2.png/v1/fill/w_499,h_68,al_c,q_85,enc_auto/64c52b_1a9016f54c8e4035af555cf82356d907~mv2.png)
![](https://static.wixstatic.com/media/64c52b_b42537c754bc40e58a64500dfe759ead~mv2.png/v1/fill/w_535,h_67,al_c,q_85,enc_auto/64c52b_b42537c754bc40e58a64500dfe759ead~mv2.png)
![](https://static.wixstatic.com/media/64c52b_9519890d1bcf416f8ffd472e4de45028~mv2.png/v1/fill/w_279,h_67,al_c,q_85,enc_auto/64c52b_9519890d1bcf416f8ffd472e4de45028~mv2.png)
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5. Summary