اندازه هاي اطلاع جنسن-فيشر و جنسن- χ2براي توزيع هاي آميخته متناهي

Jensen-Fisher and Jensen-χ2 information measures for finite mixture distributions

ﭼﮑﯿﺪه: در اﯾﻦ ﻣﻘﺎﻟﻪ، اﺑﺘﺪا ﺑﺎ در ﻧﻈﺮ ﮔﺮﻓﺘﻦ اﻃﻼع ﻓﯿﺸﺮ ﻧﻮع ﭘﺎراﻣﺘﺮي، ﯾﮏ اﻧﺪازه اﻃﻼع ﺟﺪﯾﺪ ﺑﺮ اﺳﺎس ﻧﺎﻣﺴﺎوي ﺟﻨﺴﻦ ﻣﻌﺮﻓﯽ ﻣﯽ ﮐﻨﯿﻢ. ﺳﭙﺲ ﻣﺎﺗﺮﯾﺲ اﻃﻼع ﻓﯿﺸﺮ و اﻃﻼع ﺟﻨﺴﻦ-ﻓﯿﺸﺮ را ﺑﺮاي ﯾﮏ ﺗﻮزﯾﻊ آﻣﯿﺨﺘﻪ ﻣﺘﻨﺎﻫﯽ از ﺗﻮاﺑﻊ ﭼﮕﺎﻟﯽ اﺣﺘﻤﺎل ﻣﻄﺎﻟﻌﻪ ﻣﯽ ﺷﻮد. ﻫﻤﭽﻨﯿﻦ ﯾﮏ ﻣﻌﯿﺎر اﻃﻼع دﯾﮕﺮ ﺗﺤﺖ ﻋﻨﻮان ﺟﻨﺴﻦ-χ2 ﺑﺮ اﺳﺎس ﯾﮏ ﺗﻮزﯾﻊ آﻣﯿﺨﺘﻪ از ﺗﻮاﺑﻊ ﭼﮕﺎﻟﯽ اﺣﺘﻤﺎل ﻣﻌﺮﻓﯽ ﻣﯽ ﺷﻮد. ﺗﻌﻤﯿﻢ ﻫﺎﯾﯽ از دو اﻧﺪازه اﻃﻼع ﺟﻨﺴﻦ-ﻓﯿﺸﺮ و ﺟﻨﺴﻦ-χ2 ﺑﺮ اﺳﺎسm ﺗﺎﺑﻊ ﭼﮕﺎﻟﯽ اﺣﺘﻤﺎل اراﺋﻪ و ارﺗﺒﺎط ﺑﯿﻦ اﯾﻦ دو ﻣﻌﯿﺎر اﻃﻼع ﺟﺪﯾﺪ و ﻫﻤﭽﻨﯿﻦ ارﺗﺒﺎط ﺑﯿﻦ اﻃﻼع ﺟﻨﺴﻦ-ﻓﯿﺸﺮ ﺑﺎ ﺑﺮﺧﯽ از ﻣﻌﯿﺎرﻫﺎي اﻃﻼع ﺷﻨﺎﺧﺘﻪ ﺷﺪه از ﻗﺒﯿﻞ اﻃﻼع ﺟﻨﺴﻦ-ﺷﺎﻧﻮن و اﻃﻼع ﺟﻔﺮﯾﺰ ﻣﻄﺎﻟﻌﻪ ﻣﯽ ﺷﻮد.

In this paper, first, considering Fisher information of parametric type, we introduce a new information measure based on Jensen inequality. Then, the Fisher information matrix and Jensen-Fisher are studied or a finite mixture distribution of probability density functions. Further, another information criterion is introduced as Jensen-χ2 based on a mixture of probability density functions. Generalizations of Jensen-Fisher and Jensen-χ2 information measures are presented based on m probability density functions and the relationship between these two new information criteria as well as the relationship between Jensen-Fisher information and some known information criteria such as Jensen-Shannon and Jeffreys information measures are studied.