Stepper

Four ratings, two groups, one step at a time

The explainer works EM through by hand on just four movie ratings, [2, 3, 7, 8]. Here you can replay it: each E-step works out how likely each rating is to belong to each group, and each M-step redraws the groups from those probabilities.

Starting guess

μ = 2.5 / 7.5, σ = 0.5, π = 0.5. Already at the answer, so EM stops at once.

Group 1
Group 2
Mixture

bars under each rating: γ₁ (teal) vs γ₂ (coral)

start

Initial guess (iteration 0)

Show the explainer's hand-worked numbers beside the exact ones

We have four ratings, [2, 3, 7, 8], and suspect two kinds of user behind them. EM needs a starting guess for each group's average (μ), spread (σ) and share of users (π). It does not have to be good; the steps fix it.

group	π	μ	σ
	guess
1	0.5000	2.5000	0.5000
2	0.5000	7.5000	0.5000

Press play, or step with the arrow buttons.

Explain this iterationoptional · AI · your own key

Sends this iteration's numbers (parameters before and after, responsibilities, log-likelihood, stopping rule) and a fixed description of the page and data set, with no personal data, to Anthropic, from your browser. What is sent

Step to an iteration first: the starting guess has nothing to explain yet.

Log-likelihood

-5.6758

\ell(\theta) = \sum_i \log\big(\pi_1 f(x_i \mid \mu_1, \sigma_1) + \pi_2 f(x_i \mid \mu_2, \sigma_2)\big)

\begin{aligned} \ell(\theta) = \textstyle\sum_i \log\big(&\pi_1 f(x_i \mid \mu_1, \sigma_1) \\ &+ \pi_2 f(x_i \mid \mu_2, \sigma_2)\big) \end{aligned}

What the hand-worked numbers got wrong (and right)

The explainer's worked example is the clearest part of it, so it is worth being exact about it. The density values it plugs in are off, but the story they tell is right: ratings 2 and 3 land in group 1 with near certainty, 7 and 8 in group 2, and the first M-step returns μ = 2.5 and 7.5 with σ = 0.5 and π = 0.5. In fact the starting guess was already a fixed point, so the second iteration changes nothing, exactly as the explainer says.

The original text is kept unchanged in original/em-explainer.md. The Maths page lists every correction in context.

Corrections to the worked example
quantity	as written	exact
normal_pdf(2; 2.5, 0.5)2 is exactly one standard deviation below 2.5, so the density is φ(1)/0.5.	0.8	0.4839
normal_pdf(3; 2.5, 0.5)3 is also one standard deviation from 2.5, so it must equal the density at 2.	0.6	0.4839
normal_pdf(3; 7.5, 0.5)3 is nine standard deviations from 7.5; the density is about 2 × 10⁻¹⁸.	0.0001	2.06 × 10⁻¹⁸
normal_pdf(2; 7.5, 0.5)Eleven standard deviations away; the density is about 4 × 10⁻²⁷.	0.0001	4.24 × 10⁻²⁷