#Expected value of xbar how to#

Maximization works if the function being maximized is a concave-downward function. This optimization technique is called Maximum Likelihood Estimation. In other words, we want to maximize the product of the Likelihoods в„’(О» | y_i) for all observed y_i in the training data set.

Need for concaveness of objective function: Often, we want to find the parameter vector Оё which will maximize the joint probability of observing the entire training data set.

Instead of differentiating the raw Likelihood function в„’(О» | y_i), it is often convenient to differentiate the natural logarithm of the Likelihood function, for the following reasons: О» of the likelihood в„’(О»|y) (Image by Author). The absolute value of the partial derivative w.r.t. It is read off as the Likelihood of Оё for a given observed value of the random variable y. The Y-axis depicts whatвЂ™s known as the Likelihood function of О», denoted by the stylized L, namely: в„’(О»|y), or in general в„’( Оё | y ) for some parameter Оё of the probability distribution of y. In case of the second plot shown below, the X-axis of the plot shows the parameter О». In the previous plot, the X-axis shows the observed value of the random variable y and the Y-axis shows the probability (or probability density in case of continuous y). The differences lie in what the axes of the two plot show. О» shown below is similar to the previous one in its shape. This is the sort of question that underlies the concept of the Likelihood function. Here, we are asking the question: what is the probability of observing 10 events in some unit time interval, given different values of the mean rate s О»? We fix y at 10 and allow the mean rate О» to vary from 0 to в€ћ. Now, suppose we make a single observation of y=10. for О»=16 events per unit time (Image by Author) The concept of Likelihood

#Expected value of xbar how to#

And that is exactly what Fisher information allows us to measure in the quantifiable manner that mathematics is especially good at!Д«efore we understand how to measure Fisher information contained in the random variable y, letвЂ™s once again look at the formula for the Poisson probability: It would be great to be able to know how much information is contained in y about the true but unknown mean event rate О»_0 of the population of y values. Now, suppose you observe a certain number of patients walking into the ER during a particular hour.

The best we can do is to estimate О»_0 using the sample mean rate О». Secondly, even if we assumed (correctly) that y is Poisson distributed, we would not know the value of the true mean event rate О»_0 of the population of y values.Based on the nature of the data (which in our example, non-negative counts based event data), we are assuming that y is Poisson distributed. All we have is the data sample of a few hundred events. We donвЂ™t know (and we would never know) what is the true probability distribution of y.

The Probability Mass Function of the Poisson distributed discrete random variable y (Image by Author)ДЄt this point, we need to take a step back and note the following two things: