Learn about mixture models and the EM algorithm
- Motivation and examples: it's frequent to collect heterogeneous data, ie. for which we suspect that they come from several clusters. For instance, we measure the height of individuals without recording their gender, we measure the levels of expression of a gene in several individuals without recording which ones are healthy and which ones are sick, etc.
- Data: we have N observations, noted [math]\displaystyle{ X = (x_1, x_2, ..., x_N) }[/math]. For the moment, we suppose that each observation [math]\displaystyle{ x_i }[/math] is univariate, ie. each corresponds to only one number.
- Hypotheses and aim: let's assume that the data are heterogeneous and that they can be partitioned into [math]\displaystyle{ K }[/math] clusters (see examples above). This means that a subset of the observations come from cluster [math]\displaystyle{ k=1 }[/math], another subset come from cluster [math]\displaystyle{ k=2 }[/math], and so on.
- Model: technically, we say that the observations were generated by a family of density functions. The density of all the observations is thus a mixture of densities, one per cluster. In our case, we will assume that each cluster [math]\displaystyle{ k }[/math] corresponds to a Normal distribution of mean [math]\displaystyle{ \mu_k }[/math] and standard deviation [math]\displaystyle{ \sigma_k }[/math]. Moreover, as we don't know for sure from which cluster a given observation comes from, we define the mixture probability [math]\displaystyle{ w_k }[/math] to be the probability that any given observation comes from cluster [math]\displaystyle{ k }[/math]. As a result, we have the following list of parameters: [math]\displaystyle{ \theta=(w_1,...,w_K,\mu_1,...\mu_K,\sigma_1,...,\sigma_K }[/math]. Finally, for a given observation [math]\displaystyle{ x_i }[/math], we can write the model [math]\displaystyle{ f(x_i/\theta) = \sum_{k=1}^{K} w_k g(x_i/\mu_k,\sigma_k) }[/math] , wth [math]\displaystyle{ g }[/math] being the Normal distribution [math]\displaystyle{ g(x_i/\mu_k,\sigma_k) = \frac{1}{\sqrt{2\pi} \sigma_k} \exp^{-\frac{1}{2}(\frac{x_i - \mu_k}{\sigma_k})^2} }[/math]
- Likelihood: this corresponds to the probability of obtaining the data given the parameters: [math]\displaystyle{ L(\theta) = P(X/\theta) }[/math]. We assume that the observations are independent, ie. they were generated independently, whether they are from the same cluster or not. Therefore we can write: [math]\displaystyle{ L(\theta) = \prod_{i=1}^N f(x_i/\theta) }[/math].
- Estimation: now we want to find the values of the parameters that maximize the likelihood. This reduces to (i) differentiating the likelihood with respect to each parameter, and then (ii) finding the value at which each partial derivative is zero. Instead of maximizing the likelihood, we maximize its logarithm, noted [math]\displaystyle{ l(\theta) }[/math]. It gives the same solution because the log is monotonically increasing, but it's easier to derive the log-likelihood than the likelihood. Here is the whole formula:
[math]\displaystyle{ l(\theta) = \sum_{i=1}^N log(f(x_i/\theta)) = \sum_{i=1}^N log( \sum_{k=1}^{K} w_k \frac{1}{\sqrt{2\pi} \sigma_k} \exp^{-\frac{1}{2}(\frac{x_i - \mu_k}{\sigma_k})^2}) }[/math]
- Latent variables: here it's worth noting that, although everything seems fine, a big information is lacking, we aim at finding the parameters defining the mixture but we don't know from which cluster each observation is coming... That's why we need to introduce the following N latent variables [math]\displaystyle{ Z_1,...,Z_i,...,Z_N }[/math], one for each observation, such that [math]\displaystyle{ Z_i=k }[/math] means that [math]\displaystyle{ x_i }[/math] belongs to cluster [math]\displaystyle{ k }[/math]. Thanks to this, we can reinterpret the mixing probabilities: [math]\displaystyle{ w_k = P(Z_i=k/\theta) }[/math]. Moreover, we can now define the membership probabilities, one for each observation: [math]\displaystyle{ P(Z_i=k/x_i,\theta) = \frac{w_k g(x_i/\mu_k,\sigma_k)}{\sum_{l=1}^K w_l g(x_i/\mu_l,\sigma_l)} }[/math]. We will note these membership probabilities [math]\displaystyle{ p(k/i) }[/math] as they will have a big role in the EM algorithm below. Indeed, we don't know the values taken by the latent variables, so we will have to infer their probabilities from the data.
- Technical details: a few important rules are required, but only from a high-school level in maths (see here). Let's start by finding the maximum-likelihood estimates of the mean of each cluster:
[math]\displaystyle{ \frac{\partial l(\theta)}{\partial \mu_k} = \sum_{i=1}^N \frac{1}{f(x_i/\theta)} \frac{\partial f(x_i/\theta)}{\partial \mu_k} }[/math]
As we derive with respect to [math]\displaystyle{ \mu_k }[/math], all the others means [math]\displaystyle{ \mu_l }[/math] with [math]\displaystyle{ l \ne k }[/math] are constant, and thus disappear:
[math]\displaystyle{ \frac{\partial f(x_i/\theta)}{\partial \mu_k} = w_k \frac{\partial g(x_i/\mu_k,\sigma_k)}{\partial \mu_k} }[/math]
And finally:
[math]\displaystyle{ \frac{\partial g(x_i/\mu_k,\sigma_k)}{\partial \mu_k} = \frac{\mu_k - x_i}{\sigma_k^2} g(x_i/\mu_k,\sigma_k) }[/math]
Once we put all together, we end up with:
[math]\displaystyle{ \frac{\partial l(\theta)}{\partial \mu_k} = \sum_{i=1}^N \frac{1}{\sigma^2} \frac{w_k g(x_i/\mu_k,\sigma_k)}{\sum_{l=1}^K w_l g(x_i/\mu_l,\sigma_l)} (\mu_k - x_i) }[/math]
By convention, we note [math]\displaystyle{ \hat{\mu_k} }[/math] the maximum-likelihood estimate of [math]\displaystyle{ \mu_k }[/math]:
[math]\displaystyle{ \frac{\partial l(\theta)}{\partial \mu_k}_{\mu_k=\hat{\mu_k}} = 0 }[/math]
Therefore, we finally obtain:
[math]\displaystyle{ \hat{\mu_k} = \frac{\sum_{i=1}^N p(k/i) x_i}{\sum_{i=1}^N p(k/i)} }[/math]
By doing the same kind of algebra, we also obtain the ML estimates for the standard deviation of each cluster:
[math]\displaystyle{ \hat{\sigma_k} = \sqrt{\frac{\sum_{i=1}^N p(k/i) (x_i - \mu_k)^2}{\sum_{i=1}^N p(k/i)}} }[/math]
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