Task 5. Bayesian Information Criterion

The Bayesian Information Criterion is defined as:

$//<![CDATA[ \begin{array}{l}BIC = kln(N) - 2ln(\hat{L})\end{array} //]]>$

where $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ is the number of parameters estimated by the probability model, $//<![CDATA[ \begin{array}{l}N\end{array} //]]>$ is the number of data points, and $//<![CDATA[ \begin{array}{l}\hat{L}\end{array} //]]>$ is the maximized value of the likelihood function of the probability model i.e. $//<![CDATA[ \begin{array}{l}\hat{L} = p(x|\hat{\theta}, M)\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}\hat{\theta}\end{array} //]]>$ are the parameter values that maximize the likelihood function, and $//<![CDATA[ \begin{array}{l}M\end{array} //]]>$ is the probability model. 

The log10-normal distribution is described by two parameters: mean and standard deviation. Therefore, $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ = 2.

In this case, the model parameters that optimize the likelihood function are the same as those computed using Standard Product Moments. Therefore, the maximized likelihood function is simply equal to the PDF of the normal distribution with mean of 5.023 and standard deviation of 0.230.

• Copy the sheet titled Data and name the new sheet BIC.

• Compute the PDF, $//<![CDATA[ \begin{array}{l}f_Q\end{array} //]]>$, of Q where random variable Q (discharge) follows a log10-normal distribution. To find the PDF of Q, differentiate the CDF of Y = log10(Q) and use the change of variable technique. The PDF of discharge Q is shown below and the derivation is shown in the expandable section:
  $//<![CDATA[ \begin{array}{l}f_Q(q) = \frac{f_Y[log_1_0(q)]}{qln(10)}\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}f_Y\end{array} //]]>$ is the PDF of the normal distribution 

  Derivation of PDF

  $//<![CDATA[ \begin{array}{l}Y = log_1_0(Q)\end{array} //]]>$
  $//<![CDATA[ \begin{array}{l}Q = 10^Y\end{array} //]]>$
  $//<![CDATA[ \begin{array}{l}P(Q \le q) = P(10^Y \le q) = P[Y \le log_1_0(q)] = F_Y[log_1_0(q)]\end{array} //]]>$
  $//<![CDATA[ \begin{array}{l}\frac{\delta}{\delta x} F_Y[log_1_0(q)] = f_Y[log_1_0(q)]\frac{1}{qln(10)}\end{array} //]]>$
  Therefore, the PDF of discharge Q is computed as:
  $//<![CDATA[ \begin{array}{l}f_Q(q) = \frac{f_Y[log_1_0(q)]}{qln(10)}\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}f_Y\end{array} //]]>$ is the PDF of the normal distribution 

• Compute the natural log of the likelihood function.
• Compute the BIC using the equation provided above.

Question: What is the computed Bayesian Information Criterion, BIC?

If time allows, continue to Task 6. Akaike Information Criterion.