A simple baysian question on likelihood

[ghostyc]

Hi all,

In this question, I found that

\Pr(X_i|\theta)=\frac{\exp(\theta x_i)}{1+\exp(\theta)}

and I carry on with the likelihood being \frac{\exp(\theta \sum x_i)}{(1+\exp(\theta))^n}

and so s=\sum x_i = Tn

I need some help with part (c).

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How do I start with this question then?

I couldn't get a general expression for \Pr(X_i|\theta).


How do I generally deal with these "piecewise probability density function"?

Thanks!

 

[Stephen Tashi]

Suppose that X_1,...,X_n are independent and identically distributed conditional on a random parameter \theta > 0 such that
P(X_i = 1|\theta) = \frac{e^\theta}{1+e^\theta} and P(X_i = 0| \theta) = 1 - P(X_i=1| \theta).

A normal prior for \theta is taken with mean \mu and variance \sigma^2.

(a) Show that the liklihood function is of this type

l(\theta) = \frac{e^{\theta S}}{(1 + e^\theta)^n} = \big{(}\frac{e^{\theta T}}{1 + e^\theta}\big{)}^n and find S and T in terms of \{X_1,X_2,...X_n\}.

(b) Hence, write down the posterior distribution for \theta.

(c) For any 0 < t < 1 and \theta > 0, show that

\frac{e^{\theta t }}{1 + e^\theta} < t^t(1-t)^{1-t} .

In this question, I found that

\Pr(X_i|\theta)=\frac{\exp(\theta x_i)}{1+\exp(\theta)}

and I carry on with the likelihood being \frac{\exp(\theta \sum x_i)}{(1+\exp(\theta))^n}

and so s=\sum x_i = Tn

I need some help with part (c).

One thought that comes to mind is to use calculus to see if the function
f(\theta) = (\theta)^S ( 1 - \theta)^{n-S} takes its maximum value when \theta = S/N. If so, then (f(\theta))^{1/n} also takes its maximum value there.

 

[Stephen Tashi]

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Suppose that X_1,...,X_n are independent and identically distributed conditional on a random parameter \theta > 0 such that
for each i = 1,2,...n

P(X_i = -1|\theta) = \theta} and P(X_i = 1| \theta) = 1 - \theta.

(a) Show that the liklihood function is given by the form

l(\theta) = \theta^{n/2 - S} (1-\theta)^{n/2 + S} and find S in terms of \{X_1,X_2,...X_n\}.
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let k [/tex] be the number of the X_i that are +1 and let S = n/2 - k<br /> <br /> Isn&#039;t &quot;discrete&quot; a better term than &quot;piecewise&quot;?