# “Three unblinded mice” from Andrew Gelman’s blog

“Three unblinded mice” from Andrew Gelman’s blog

I really like the phrase “pre-scientific intuitions”. Also this: “OK, sure, sure, everybody knows about statistics and p-values and all that, but my impression is that researchers see these methods as a way to prove that an effect is real. That is, statistics is seen, not as a way to model variation, but as a way to remove uncertainty.”

# Converting between decimal and unary in Matlab

Decimal to unary:

[1:4]'==3   % returns [0 0 1 0 ]

Unary to decimal:

max([0 0 1 0] .* (1:4))   % returns 3

# Fisher information, score function, oh my!

Ever wondered why $\text{Var}_{\theta}(\ell'(\theta)) = I_n(\theta)$ and $\text{Var}_{\theta}(\hat{\theta}) \approx \frac{1}{I_n(\theta)}$?

This is an excellent intuitive explanation:

http://math.stackexchange.com/a/265933/105416

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# Confidence Intervals from Pivots

This is an example of using a pivot to find a confidence interval. $X_1,...,X_n \sim \text{Uniform}(0,\theta).$

1. Find a pivot:

Let $Q=X_{(n)}/\theta$.

2. Find its distribution: $P(Q \le t)= P(X_i \le t\theta)^n = t^n$.

3. Find an expression involving an upper and lower bound on the pivot: $P(a \le Q \le b) = b^n-a^n$ This implies $P(a \le Q \le 1) = 1-a^n$.

4. Substitute the expression for the pivot from Step 1, and set the RHS to $1-\alpha$. $P(a \le X_{(n)}/\theta \le 1)=1-a^n$ $P(1/a \ge \theta/X_{(n)} \ge 1) = 1-a^n$ $P( X_{(n)} \le \theta \le \frac{X_{(n)}}{a} ) = 1-a^n$

Let $1-\alpha = 1-a^n$. Then $a=\alpha^{1/n}$. $P(X_{(n)} \le \theta \le \frac{X_{(n)}}{\alpha^{1/n}})=1-\alpha$

This gives us $[X_{(n)},\frac{X_{(n)}}{\alpha^{1/n}}]$ as a $1-\alpha$ CI for $\theta$.

# Hello world $\int_0^1 f(x) \, \mathrm{d} x$