graph LR idea_054dd59e_cffa_4d1b_8d3b_0ea14b8fdcac["Joint Probability"] idea_0d819784_184d_4fff_adac_c952ef62af72["Centroid"] idea_0f2d8b11_0653_4925_95b8_fec4fe43ab43["Determinant"] idea_1b0e4c9b_3ab8_467e_8f52_2054c5cf5113["E-Step of the EM Algorithm"] idea_1bfd7c38_cfcd_4e36_94e5_d83984cdc219["Gaussian Mixture Model"] idea_1c96b953_b36c_4713_bb5a_21fb5ba46082["Categorical Distribution"] idea_212aa94a_0198_40b5_be5c_4743aa013ff6["Selecting the Number of Mixture Components"] idea_2178b825_dc09_4f78_86c3_913b36ca0d8f["Responsibilities in Mixture Models"] idea_26431081_04bf_42df_b5f6_812039f06d73["Covariance Matrix"] idea_39c329f1_78f5_4b4b_8284_59779478a47b["Overfitting"] idea_3db74e87_2793_4bad_bc9a_24bdef5ee432["Maximum Likelihood Estimation"] idea_4519e78f_18ce_440d_adbb_a9da35bbfe55["Covariance Structures in Gaussian Mixtures"] idea_4f519a78_3938_4115_ab39_09ad3009b1bc["Soft Versus Hard Cluster Assignment"] idea_51c2bd68_8ebc_4e38_b01f_8b1e2479ba62["Euclidean Distance"] idea_5380fb69_fc05_4466_b5b2_544579495466["K-Means Clustering"] idea_541f63b5_10dd_4219_a721_16d76f001cee["Latent Variable Formulation of Mixture Models"] idea_577ba734_8860_46df_818c_2c7836165040["Marginal Probability"] idea_5b5f30be_b5fb_459a_aa68_6878f09d9c0b["Arithmetic Mean"] idea_5bb5b54b_7be2_4efb_ba45_94f9cc0eceb2["Singular Covariance Collapse"] idea_6a8fd877_6ffd_4757_8e59_5a8c42cb6af3["M-Step of the EM Algorithm"] idea_6b56ae1c_6adb_4ec4_937a_d0de259cd632["Bayesian Information Criterion"] idea_71499bd4_c040_4f32_89b8_1351422dc90e["Convex Function"] idea_71a538b8_7978_41bf_b566_bccaa1ac96b9["Probability Density Function"] idea_721c5fb3_d6fb_4bca_9cc6_fa9187259f9d["Local Optima in EM"] idea_7726a79f_7c09_4a11_abe0_37ae7cdbd73c["Lagrange Multipliers"] idea_7f8235b3_3c84_4c34_9289_ca52cfef9b91["Bayes' Theorem"] idea_81a1ab41_cd27_473f_9d15_b8538e8f51e4["Density Estimation with Mixture Models"] idea_84aef63a_28c9_4739_93c2_c7765cbf88ba["Log-Likelihood Function"] idea_866205fd_d22b_4057_b854_265697f8c57f["Positive Definite Matrix"] idea_8e514a7c_e0e2_4e6f_ad19_9448958ecc60["K-Means Seeding for EM Initialization"] idea_988d5ea7_f91c_4c6a_9e8e_94fc8ddfe21c["Jensen's Inequality"] idea_a3171efc_56ba_46e4_96f6_23fe9e03c858["Akaike Information Criterion"] idea_a776aecf_8678_4f81_b665_8eba7d264040["Conditional Probability"] idea_ab4a88e0_45ba_41b4_b0ca_7300362f5609["Posterior Probability"] idea_ac3c5700_7436_4e40_b665_326fc6275144["Mel-Frequency Cepstral Coefficients"] idea_ac9190b2_ae3f_463a_bd41_cd8a752b27c2["Expected Value"] idea_b75faf6e_5848_4c74_a63c_cac9a8841ffd["Gaussian (Normal) Distribution"] idea_c8011b6e_c4bc_4e3c_b4dd_61b734413166["Universal Background Model"] idea_cd3349fe_9766_4eda_92ce_057293bd79c0["Multivariate Gaussian Distribution"] idea_ead8b062_cec4_47c3_acb4_62852734b4bc["Monotonic Likelihood Ascent of EM"] idea_f4c64527_f83e_47e4_98d5_3ffc1def9fc8["Likelihood Function"] idea_f93b26cc_bad2_4e37_b30d_042b294c2578["Kullback–Leibler Divergence"] idea_fcf58943_8045_4a38_b633_55d5b1a2a754["Mixture Weights"] idea_ff79ed67_f58f_471a_bf80_c1ea8d58652d["Matrix Inverse"] idea_7f8235b3_3c84_4c34_9289_ca52cfef9b91 --> idea_c8011b6e_c4bc_4e3c_b4dd_61b734413166 idea_71a538b8_7978_41bf_b566_bccaa1ac96b9 --> idea_1bfd7c38_cfcd_4e36_94e5_d83984cdc219 idea_0f2d8b11_0653_4925_95b8_fec4fe43ab43 --> idea_5bb5b54b_7be2_4efb_ba45_94f9cc0eceb2 idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 --> idea_5bb5b54b_7be2_4efb_ba45_94f9cc0eceb2 idea_577ba734_8860_46df_818c_2c7836165040 --> idea_fcf58943_8045_4a38_b633_55d5b1a2a754 idea_3db74e87_2793_4bad_bc9a_24bdef5ee432 --> idea_6a8fd877_6ffd_4757_8e59_5a8c42cb6af3 idea_84aef63a_28c9_4739_93c2_c7765cbf88ba --> idea_ead8b062_cec4_47c3_acb4_62852734b4bc idea_ff79ed67_f58f_471a_bf80_c1ea8d58652d --> idea_cd3349fe_9766_4eda_92ce_057293bd79c0 idea_0f2d8b11_0653_4925_95b8_fec4fe43ab43 --> idea_cd3349fe_9766_4eda_92ce_057293bd79c0 idea_26431081_04bf_42df_b5f6_812039f06d73 --> idea_4519e78f_18ce_440d_adbb_a9da35bbfe55 idea_26431081_04bf_42df_b5f6_812039f06d73 --> idea_6a8fd877_6ffd_4757_8e59_5a8c42cb6af3 idea_7f8235b3_3c84_4c34_9289_ca52cfef9b91 --> idea_2178b825_dc09_4f78_86c3_913b36ca0d8f idea_ac9190b2_ae3f_463a_bd41_cd8a752b27c2 --> idea_1b0e4c9b_3ab8_467e_8f52_2054c5cf5113 idea_ab4a88e0_45ba_41b4_b0ca_7300362f5609 --> idea_1b0e4c9b_3ab8_467e_8f52_2054c5cf5113 idea_1c96b953_b36c_4713_bb5a_21fb5ba46082 --> idea_fcf58943_8045_4a38_b633_55d5b1a2a754 idea_b75faf6e_5848_4c74_a63c_cac9a8841ffd --> idea_1bfd7c38_cfcd_4e36_94e5_d83984cdc219 idea_26431081_04bf_42df_b5f6_812039f06d73 --> idea_cd3349fe_9766_4eda_92ce_057293bd79c0 idea_a776aecf_8678_4f81_b665_8eba7d264040 --> idea_2178b825_dc09_4f78_86c3_913b36ca0d8f idea_84aef63a_28c9_4739_93c2_c7765cbf88ba --> idea_1b0e4c9b_3ab8_467e_8f52_2054c5cf5113 idea_ac3c5700_7436_4e40_b665_326fc6275144 --> idea_c8011b6e_c4bc_4e3c_b4dd_61b734413166 idea_51c2bd68_8ebc_4e38_b01f_8b1e2479ba62 --> 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idea_8e514a7c_e0e2_4e6f_ad19_9448958ecc60 idea_f4c64527_f83e_47e4_98d5_3ffc1def9fc8 --> idea_721c5fb3_d6fb_4bca_9cc6_fa9187259f9d idea_1c96b953_b36c_4713_bb5a_21fb5ba46082 --> idea_541f63b5_10dd_4219_a721_16d76f001cee idea_7726a79f_7c09_4a11_abe0_37ae7cdbd73c --> idea_6a8fd877_6ffd_4757_8e59_5a8c42cb6af3 idea_71a538b8_7978_41bf_b566_bccaa1ac96b9 --> idea_81a1ab41_cd27_473f_9d15_b8538e8f51e4 idea_866205fd_d22b_4057_b854_265697f8c57f --> idea_5bb5b54b_7be2_4efb_ba45_94f9cc0eceb2 idea_6b56ae1c_6adb_4ec4_937a_d0de259cd632 --> idea_212aa94a_0198_40b5_be5c_4743aa013ff6 idea_b75faf6e_5848_4c74_a63c_cac9a8841ffd --> idea_cd3349fe_9766_4eda_92ce_057293bd79c0 idea_ab4a88e0_45ba_41b4_b0ca_7300362f5609 --> idea_2178b825_dc09_4f78_86c3_913b36ca0d8f idea_866205fd_d22b_4057_b854_265697f8c57f --> idea_4519e78f_18ce_440d_adbb_a9da35bbfe55 idea_f93b26cc_bad2_4e37_b30d_042b294c2578 --> idea_ead8b062_cec4_47c3_acb4_62852734b4bc style idea_8e514a7c_e0e2_4e6f_ad19_9448958ecc60 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