I got stuck in a some step ≥ [ ∑ z P ( z | X , θ n ) ⋅ ln ⁡ ( P ( X | z , θ ) P ( z | θ ) P ( z | X , θ n ) ) ] − ln ⁡ ( P ( X | θ n ) ) which in the next line results in = ∑ z P ( z | X , θ n ) ⋅ ln ⁡ ( P ( X | z , θ ) P ( z | θ ) P ( z | X , θ n ) P ( X | θ n ) ) What happened in between those lines?

Question

I got stuck in a some step   ≥            [            ∑    z    P  (  z      |    X  ,      θ    n    )  ⋅  ln  ⁡            (                  P      (      X              |            z      ,      θ      )      P      (      z              |            θ      )              P      (      z              |            X      ,              θ        n            )                  )                  ]        −  ln  ⁡            (        P  (  X      |        θ    n    )            )      which in the next line results in  =      ∑    z    P  (  z      |    X  ,      θ    n    )  ⋅  ln  ⁡            (                  P      (      X              |            z      ,      θ      )      P      (      z              |            θ      )              P      (      z              |            X      ,              θ        n            )      P      (      X              |                    θ        n            )                  )      What happened in between those lines?

laure6237ma · Accepted Answer

Note that       ∑    z    P  (  z      |    X  ,      θ    n    )  =  1. This way,                                          [                                    ∑          z                P        (        z                  |                X        ,                  θ          n                )        ⋅                    ln        ⁡                  (                                    P              (              X                              |                            z              ,              θ              )              P              (              z                              |                            θ              )                                      P              (              z                              |                            X              ,                              θ                n                            )                                )                                      ]                          −        ln        ⁡        P        (        X                  |                          θ          n                )                                          =                              [                                    ∑          z                P        (        z                  |                X        ,                  θ          n                )        ⋅        ln        ⁡                  (                                    P              (              X                              |                            z              ,              θ              )              P              (              z                              |                            θ              )                                      P              (              z                              |                            X              ,                              θ                n                            )                                )                                      ]                          −        ln        ⁡        P        (        X                  |                          θ          n                )                  ∑          z                P        (        z                  |                X        ,                  θ          n                )                                          =                              [                                    ∑          z                P        (        z                  |                X        ,                  θ          n                )        ⋅        ln        ⁡                  (                                    P              (              X                              |                            z              ,              θ              )              P              (              z                              |                            θ              )                                      P              (              z                              |                            X              ,                              θ                n                            )                                )                                      ]                          −                  ∑          z                P        (        z                  |                X        ,                  θ          n                )        ln        ⁡        P        (        X                  |                          θ          n                )                                          =                  ∑          z                P        (        z                  |                X        ,                  θ          n                )        ⋅                  (          ln          ⁡                      (                                          P                (                X                                  |                                z                ,                θ                )                P                (                z                                  |                                θ                )                                            P                (                z                                  |                                X                ,                                  θ                  n                                )                                      )                    −          ln          ⁡          P          (          X                      |                                θ            n                    )          )                                                  =                  ∑          z                P        (        z                  |                X        ,                  θ          n                )        ⋅                  (          ln          ⁡                      (                                          P                (                X                                  |                                z                ,                θ                )                P                (                z                                  |                                θ                )                                            P                (                z                                  |                                X                ,                                  θ                  n                                )                P                (                X                                  |                                                  θ                  n                                )                                      )                    )                ,            where the last equality is the usual log property:   ln  ⁡  (  a  )  −  ln  ⁡  (  b  )  =  l  n  (  a      /    b  ).

I got stuck in a some step ≥ <mstyle scriptlevel="0"> <mrow class="MJ

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