/***      Author: Tapas Kanungo, kanungo@cfar.umd.edu**      Date:   15 December 1997**      File:   backward.c**      Purpose: Backward algorithm for computing the probabilty**              of observing a sequence given a HMM model parameter.**      Organization: University of Maryland****      $Id: backward.c,v 1.3 1998/02/23 07:56:05 kanungo Exp kanungo $*/#include <stdio.h>#include "hmm.h"static char rcsid[] = "$Id: backward.c,v 1.3 1998/02/23 07:56:05 kanungo Exp kanungo $";void Backward(HMM *phmm, int T, int *O, double **beta, double *pprob){        int     i, j;   /* state indices */        int     t;      /* time index */        double sum;          /* 1. Initialization */         for (i = 1; i <= phmm->N; i++)                beta[T][i] = 1.0;         /* 2. Induction */         for (t = T - 1; t >= 1; t--) {                for (i = 1; i <= phmm->N; i++) {                        sum = 0.0;                        for (j = 1; j <= phmm->N; j++)                                sum += phmm->A[i][j] *                                        (phmm->B[j][O[t+1]])*beta[t+1][j];                        beta[t][i] = sum;                 }        }         /* 3. Termination */        *pprob = 0.0;        for (i = 1; i <= phmm->N; i++)                *pprob += beta[1][i]; }void BackwardWithScale(HMM *phmm, int T, int *O, double **beta, 	double *scale, double *pprob){        int     i, j;   /* state indices */        int     t;      /* time index */	double sum;          /* 1. Initialization */         for (i = 1; i <= phmm->N; i++)                beta[T][i] = 1.0/scale[T];          /* 2. Induction */         for (t = T - 1; t >= 1; t--) {                for (i = 1; i <= phmm->N; i++) {			sum = 0.0;                        for (j = 1; j <= phmm->N; j++)                        	sum += phmm->A[i][j] * 					(phmm->B[j][O[t+1]])*beta[t+1][j];                        beta[t][i] = sum/scale[t];                 }        } }