@@ -46,7 +46,7 @@ and is essentially the factor that we have to multiply the unbiased probability
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@@ -46,7 +46,7 @@ and is essentially the factor that we have to multiply the unbiased probability
the \f$k\f$th of our biased simulations. Obviously, these \f$w_{kj}\f$ values depend on the value that the CVs take and also on the particular trajectory that we are investigating
the \f$k\f$th of our biased simulations. Obviously, these \f$w_{kj}\f$ values depend on the value that the CVs take and also on the particular trajectory that we are investigating
all of which, remember, have different simulation biases. Finally, \f$c_k\f$ is a free parameter that ensures that, for each \f$k\f$, the biased probability is normalized.
all of which, remember, have different simulation biases. Finally, \f$c_k\f$ is a free parameter that ensures that, for each \f$k\f$, the biased probability is normalized.
We can use the equation for the probablity that was given above to find a set of values for \f$p_j\f$ that maximizes the likelihood of observing the trajectory.
We can use the equation for the probability that was given above to find a set of values for \f$p_j\f$ that maximizes the likelihood of observing the trajectory.
This constrained optimization must be performed using a set of Lagrange multipliers, \f$\lambda_k\f$, that ensure that each of the biased probability distributions
This constrained optimization must be performed using a set of Lagrange multipliers, \f$\lambda_k\f$, that ensure that each of the biased probability distributions
are normalized, that is \f$\sum_j c_kw_{kj}p_j=1\f$. Furthermore, as the problem is made easier if the quantity in the equation above is replaced by its logarithm
are normalized, that is \f$\sum_j c_kw_{kj}p_j=1\f$. Furthermore, as the problem is made easier if the quantity in the equation above is replaced by its logarithm
which can be solved by computing the \f$p_j\f$ values using the first of the two equations above with an initial guess for the \f$c_k\f$ values and by then refining
which can be solved by computing the \f$p_j\f$ values using the first of the two equations above with an initial guess for the \f$c_k\f$ values and by then refining
these \f$p_j\f$ values using the \f$c_k\f$ values that are obtained by inserting the \f$p_j\f$ values obtained into the second of the two equations above.
these \f$p_j\f$ values using the \f$c_k\f$ values that are obtained by inserting the \f$p_j\f$ values obtained into the second of the two equations above.
Notice that only \f$\sum_k t_{kj}\f$, which is the total number of configurations from all the replicas that enter the \f$j\f$th bin, enters the WHAM equations above.
Notice that only \f$\sum_k t_{kj}\f$, which is the total number of configurations from all the replicas that enter the \f$j\f$th bin, enters the WHAM equations above.
There is thus no need to record which replica generated each of the frames. One can thus simply gather the trajectories from all the replicas together at the outset.
There is thus no need to record which replica generated each of the frames. One can thus simply gather the trajectories from all the replicas together at the outset.
This observation is important as it is the basis of the binless formulation of WHAM that is implemented within PLUMED.
This observation is important as it is the basis of the binless formulation of WHAM that is implemented within PLUMED.